If you're not familiar with the しか～ない construction, please read this first.
The basic structure of the whole sentence is:
A is established only by B.
Nothing but B can make/form A.
- A is a noun phrase, 好きになるというの ("falling in love with someone")
- B is another long noun phrase, 自分の一番皮膚の薄い柔らかい場所を差し出すこと (lit. "giving the part where your skin is thinnest and softest")
So the whole sentence roughly means "To love someone, one must give/devote the thinnest and softest skin part of you body" or something like this.
To translate this naturally, I need some more contexts. In particular, I'm not sure what B is actually referring to. B may figuratively mean "showing the fragile part of your mind" or "being honest", but it may have a more sensual and explicit meaning.