Trim Paths > offset

You will never get smooth corners unless the vertices are in exactly the same position. The endpoints of two shape layer paths will not snap to each other, but they will snap to a grid or guide.  Unless the angle between both lines is 90º, you will never get a smooth corner from two lines unless you choose Round Caps. The artwork needs to be redrawn using a single path, or the points will always look strange.

The only other option you have for snapping endpoints to each other would be to use the Create Nulls from Paths/Points Follow Nulls script and snap the nulls for the endpoints to the same position. That will not solve the round cap issue. The endpoints of your paths may match up. If they do, there is no way to get a flat end to the lines because the flat ends of the lines will always be perpendicular to the path. Grouping or Merge Paths will not solve the issue. If you want flat ends to the pointed lines, redrawing the artwork is the only solution.

If you want to redraw the artwork quickly, select all of the paths, copy them, paste them into a comp-sized Illustrator document, and then use the direct selection tool (a) and drag a selection around the endpoints, right-click or use the Keyboard Shortcut Ctrl/Cmnd + j to join the vertices. When you have all of the paths joined and verified by checking the Layer Panel, you can copy and paste the path to a new shape layer in After Effects or just finish the artwork in Illustrator and import the new file as a composition retaining layer size. 

You should see the mismatched endpoints if you toggle mask and shape path visibility to on. 

Shubhi,

The Best and safest way to join is to ensure that the two end Anchor Points coincide, and then simply Cmd+J.

You can ClickDrag either path by its relevant end Anchor Point with the (normal) Selection Tool to snap to its counterpart on the other path, Smart Guides being your friends by saying anchor when you are within snapping distance.

Then you can deselect and ClickDrag over the two coinciding end Anchor Points with the Direct Selection Tool and Cmd+J.

If done in the right way, you get the option of clicking Corner rather than Smooth.

Using a less accurate way than the one described can lead to two non coincident Anchor Points which will need to be brought to together with Average or whatever, and this can lead further to unecessary issues.

Could it be the shape layer outlines causing this? 

Could you post a screenshot of the shape layer's Stroke settings.  It looks to me a little like each line is its own shape with it's own end cap.  And is there a chance that there's your original layer still turned on underneath it?  Actually could you also post a screenshot of your timeline

I've moved this from the Using the Community forum (which is the forum for issues using the forums) to the AE forum so that proper help can be offered.