"Concern" is a noun that means "worry". You use the word "concern" to talk about people's worried feelings and emotions. For example, if you get injured and someone calls to ask if you're OK, you should say:

I appreciate your concern.

Here's another example, where the speaker is talking about a business project that she's worried about:

My concern is that there may be additional costs that we haven't prepared for.

To "deepen" someone's concern means to make them more worried.

And finally, you can explain the reason that someone is worried with "over ___"

People's concern over the economy is deepening.

There's some concern over whether we'll be able to make the deadline.

The phrase "deepen (someone's) concern over (something)" is pretty formal.