How can I cleanly cut the inside of a paper on a different angle?
Using scissors, as you have pictured, would be a contributing factor here. Using a craft knife or something similar would most certainly be the go-to tool. 

It is important to try to do the cutting in one motion - to not lift the blade from the paper and, if possible, not change the pressure you are exerting on the scissors. So if you were cutting this on a mat (without scissors of course), when you need to change direction you can rotate the paper accordingly. Craft knives have fine points so turning it is not likely to damage too much. 

In the case of the specific Y, using a craft knife, I would also consider the starting point of the cuts to be the intersection shown and then cut away from that point. That approach is more case-by-case and really is dependent on your overall design. 

If you insist on using scissors, the only tip I have to offer is to try and keep cutting as you are turning the paper. Try and finish the whole cut in one motion of the scissors. That would help stop the start of a tear. Hands and the tool itself can make that hard.