This is used during anime production,but why do animators draw those and what are they for?
It looks like an inbetweening timing chart, very similar to the ones given in an example on a web page titled Timing Charts by Brian Lemay:
The chart in your question looks to be an example of what the is described in the picture above as a "halves slo-in" and "halves slo-out".
The chart shows how the inbetween frames that fill the gaps between key frames (like the one below the chart in your question) should be drawn. The chart gives two alternatives, one using 5 inbetween frames between key frames on the left, and the other using 4 inbetween frames between them.
Using the alternative on the left, the chart is telling the inbetweener to first draw frame number 5. This frame would be drawn halfway between two key frames, the one numbered 1 and 7 on the chart. The inbetweener would then draw frames 4 and 6, the former halfway between frames 1 and 5 and the later halfway between 5 and 7. Then frame 3 would be drawn, halfway between 1 and 4, and then frame 2. halfway between 1 and 3. A similar process would be followed to draw frames 9, 8, 10, 11, and 13 in that order.
Since the frames are displayed at a constant rate (probably "on twos" at 12 frames per second) the illusion of movement created by the animation would start off slow, speed up and then slow down again. The total running time for the animated sequence would be (if done on twos) 1 second.
As noted in the comments, the first two in particular are likely to be animation timings — in particular, the first two (tagged 1-7 and 7-1, though the numbers on the latter actually run from 8 to 12 with the same number of internal markings as the first) illustrate the principles of ease-out and ease-in ; motions should be slower at their start and finish than in the middle. For a good brief tutorial on this, see e.g. http://blog.digitaltutors.com/animation-body-mechanics-ease-in-and-ease-out/ — it's one of the classic principles of animation and goes back to the very earliest days of the genre.