In the past the only ways to repost to Instagram were to download photos from the platform and then post them on your own feed or to use a third-party app, which could be risky, and often meant having an ugly logo in the corner of the image if you used the free version of the software.
This is no longer necessary because Instagram has introduced its own repost option (just look for the arrow and 'repost' below an image). The option is only available if a user has enabled it. You can also share posts as a Story.