Take a sheet of paper. That's your wall. Lay it
flat on your desk. Any point on the surface of the paper or your desk top is "in
the same plane".

Pick the paper up, tear it into two pieces, and put the two pieces back on your desk. Re-align the two pieces so that your paper looks like one piece again. The tear is your crack. Your wall has a crack but both pieces are still "in the same plane".

Now for planes of movement. There are three of them, plus rotation.

Take one of the pieces of paper, and while keeping it flat on the desk, move it away from the other one slightly, so that the crack gets gradually wider. Keep the open space between the two pieces of paper equal along the full length of the crack. That's one plane of movement.

Now, put the paper back together so that it looks like one piece again. Take a pencil and draw a short straight line from one piece, across the tear, and on to the other piece. While holding one piece down against the desk so it doesn't move, take the other piece and slide it slowly along the crack, so that the pencil lines on each piece remain parallel to each other while the gap between the lines gets wider. Both pieces are still on your desk. The two pieces of paper should not have a gap between them; they are sliding relative to each other. That's your second plane of movement.

Put the paper back together again so it looks like one piece. Leave one piece on your desk. Take the other piece and raise it slowly off your desk, keeping the surface of that piece of paper parallel to the surface of the desk. Don't move it in the direction that you moved it in the first example, and don't slide it relative to the other one like you did in the second example. If you looked down on it from straight above, it would still look like one piece of paper, with one line drawn on it. If you looked at it from the side you would see two pieces of paper, one laying on the desk and one slightly above the surface of the desk. That's your third plane of movement.

None of this so far involved rotation. Let's add it now.

Put the two pieces back on the desk and align them so that they look like one piece of paper again. Grab one piece at the edge that is opposite the crack and raise that edge off the desk. There is no gap between the pieces of paper and the crack is acting like a hinge. That's one axis of rotation.

Align the two pieces of paper so it looks like it is one piece again. This time, leave both pieces flat on the desk, but take one and rotate it slightly, so that the crack becomes wide at one end and narrow at the other end. There's your second axis of rotation.

Put the paper back so it looks like one piece again. Leave one piece flat on the desk but grab the other piece on either one of the two sides that touch the side that is torn. Take the side you grabbed and raise it up off the table, while leaving the opposite side against the table. The side against the table is acting like a hinge. If put your eye down close to the desk and you looked toward the crack from the side, you would see a gap between the two pieces of paper. The gap would be wide at the edge raised off the table, and narrow at the edge still on the table. If you looked down on it from directly above, you would not be able to see any gap between the two pieces of paper. That's your third axis of rotation.

In these examples I've used the piece of paper that I kept flat on the desk the whole time, and a point on the torn edge of it, as my common frame of reference. This is because I want an easy way to describe how one piece has moved relative to the other piece. We could also pick any point on a stationary object anywhere in the room and use it to describe the motion of each piece of paper in three planes of motion and along three axes of rotation, relative to that point and relative to each other.

Pick the paper up, tear it into two pieces, and put the two pieces back on your desk. Re-align the two pieces so that your paper looks like one piece again. The tear is your crack. Your wall has a crack but both pieces are still "in the same plane".

Now for planes of movement. There are three of them, plus rotation.

Take one of the pieces of paper, and while keeping it flat on the desk, move it away from the other one slightly, so that the crack gets gradually wider. Keep the open space between the two pieces of paper equal along the full length of the crack. That's one plane of movement.

Now, put the paper back together so that it looks like one piece again. Take a pencil and draw a short straight line from one piece, across the tear, and on to the other piece. While holding one piece down against the desk so it doesn't move, take the other piece and slide it slowly along the crack, so that the pencil lines on each piece remain parallel to each other while the gap between the lines gets wider. Both pieces are still on your desk. The two pieces of paper should not have a gap between them; they are sliding relative to each other. That's your second plane of movement.

Put the paper back together again so it looks like one piece. Leave one piece on your desk. Take the other piece and raise it slowly off your desk, keeping the surface of that piece of paper parallel to the surface of the desk. Don't move it in the direction that you moved it in the first example, and don't slide it relative to the other one like you did in the second example. If you looked down on it from straight above, it would still look like one piece of paper, with one line drawn on it. If you looked at it from the side you would see two pieces of paper, one laying on the desk and one slightly above the surface of the desk. That's your third plane of movement.

None of this so far involved rotation. Let's add it now.

Put the two pieces back on the desk and align them so that they look like one piece of paper again. Grab one piece at the edge that is opposite the crack and raise that edge off the desk. There is no gap between the pieces of paper and the crack is acting like a hinge. That's one axis of rotation.

Align the two pieces of paper so it looks like it is one piece again. This time, leave both pieces flat on the desk, but take one and rotate it slightly, so that the crack becomes wide at one end and narrow at the other end. There's your second axis of rotation.

Put the paper back so it looks like one piece again. Leave one piece flat on the desk but grab the other piece on either one of the two sides that touch the side that is torn. Take the side you grabbed and raise it up off the table, while leaving the opposite side against the table. The side against the table is acting like a hinge. If put your eye down close to the desk and you looked toward the crack from the side, you would see a gap between the two pieces of paper. The gap would be wide at the edge raised off the table, and narrow at the edge still on the table. If you looked down on it from directly above, you would not be able to see any gap between the two pieces of paper. That's your third axis of rotation.

In these examples I've used the piece of paper that I kept flat on the desk the whole time, and a point on the torn edge of it, as my common frame of reference. This is because I want an easy way to describe how one piece has moved relative to the other piece. We could also pick any point on a stationary object anywhere in the room and use it to describe the motion of each piece of paper in three planes of motion and along three axes of rotation, relative to that point and relative to each other.