Thought exercise - you live in the country and one night an alien spaceship lands in your yard, and an alien emerges and engages in telepathy with you. The ship departs, leaving evidence of its landing - dents in the ground, scorch marks, etc. Maybe your family even see it too. But it wasn't tracked conveniently by satellites.
Explain how you prove it happened.
In this case you saw the aliens and you know that it has happened. You don't need to prove it. There is no need to believe in aliens either. Believing in something doesn't make it true or false.
You can explain your story to others. Perhaps people who are very interested in such matters. They will build instruments, better instruments, more instruments to detect alien vessels. You can hope that they will come back and the instruments will detect them. The story and photos will get printed. The information will be passed along. Other countries will join in on the effort of detecting them. etc etc.