What is the reasoning behind the President using many pens to sign a bill into law?

They’re Presidential souvenirs.

Here is a link to an article on the tradition and some examples.

From the article @Pied_Pfeffer links to:

The practice of using multiple pens to sign important legislation dates at least as far back as Franklin Roosevelt and is now one of our government’s frivolous little quirks, much like that oversize gavel Nancy Pelosi carried around the other day.

The rationale is fairly simple. The pen used to sign historic legislation itself becomes a historical artifact. The more pens a President uses, the more thank-you gifts he can offer to those who helped create that piece of history.

Each person who did significant work on a bill gets a pen. The pens themselves then have some significance to those people. I had this experience in the state of Washington, after working very hard on a bill—all the people who had done much work were assembled, and the pens were distributed to us.

I believe the president has chosen who will get the pens, but I did see him give one to a little boy standing by his desk. And I know the child did not work on a bill or anything. Perhaps his father did. Who knows, who cares? Not me.

It seemed like a good idea at the time….
It was a vast Right-wing conspiracy…
I am not a crook….