He has contributed to PSG for 7 years and has won various titles and can be said to have been an extraordinary player during that time. From the club's all-time top scorer to the highest paid player, Kylian Mbappe is already pinned down and he just needs to win the title he can't win with his new club later.
I think this is what Kylian Mbappe wants after being satisfied with PSG, he just needs to expand his career elsewhere and there's nothing wrong with him becoming a Real Madrid legend. But it will still be just the beginning and it will definitely be very difficult to be in a team with competition from some other great players. He is certain to win the title every season, but for individual awards it will be the most difficult thing at Real Madrid.
So far he no longer needs to prove what in Ligue 1, as far as his performances and also his contribution at PSG are huge and as you said of course Mbappe now needs to look for other proofs in other leagues, so far Mbappe has managed to prove his capacity as the best player and attacker in Ligue 1, won many championship trophies and also has the status of Ligue 1 top scorer in a row so far of course proves him It deserves to be appreciated very expensively, but of course he also still has dreams that have not been fulfilled and maybe even will not complete all the achievements he achieved if he has not been able to win the Champions League trophy and Ballon'Or during his career.
Dissatisfaction with all that he has achieved certainly made him choose to go to Madrid in order to fulfill a dream that he had difficulty getting if he stayed with PSG, currently Madrid and Mbappe are rumored to be working together but so far we have not heard any confirmation from both parties this time, but strong rumors circulating are of course most likely Mbappe will join Madrid, So I think other La Liga clubs should be wary of the attacking line owned by Madrid next season.