I am concerned/worried that: what is the function etc of 'that'?

That here is a subordinator introducing the clause I am losing weight. The idiom I am worried that makes the precise syntactic role of that clause a little muddy, but I think the simplest analysis that it marks I am losing weight as the agent of your concern:

[That I am losing weight] concerns me = I am concerned that I am losing weight.
[That I am losing weight] makes me happy = I am happy that I am losing weight.

Adjectives can have subjects just like verbs, though it may seem odd to think about it that way.

Technically speaking, the word that is a complementiser for the object complement clause "I am losing weight", but I'm not sure that will help OP understand what's going on here.

The first point to note is that it's completely optional in most contexts (including OP's example), and including it or not makes no difference to either the meaning or the level of "formality". It therefore follows that in this particular context, it doesn't really "mean" anything at all.

The second point to note is that OP's entire sentence is ambiguous - with or without that, and regardless of whether the verb is worried or concerned. The two possible meanings are:

1: I'm apprehensive because I know I'm losing weight.
2: I'm don't know if I'm losing weight, but I'm fretting about the possibility that I might be.

Choosing between those interpretations is subjective for the listener, even if we assume the speaker knows which he meant. My own feeling is that #2 is more common overall in such constructions, but you can always disambiguate by using because or in case instead of the optional that, to force meaning #1 or #2.