Translation of Loic Merel's Paper on Bounds for Elliptic Curves
Solution 1:
Googling a little, I found this expository paper: https://lmbp.uca.fr/~rebolledo/page-fichiers/07-goettingen.pdf
I haven't read it and can't vouch for its completeness, but between reading this and looking at the original paper in French you should be able to piece together any missing steps.
If you go further into arithmetic geometry, you will probably run into this problem again. It is probably worth forcing yourself to read a few papers in French (with help from Google and people who already know the math) until you are reasonably comfortable with reading mathematical French.