I'm confused about why this is subjunctive. The subjunctive is generally used with actions that may not happen. If you are in the process of starting to bring the towels that is an action that is currently happening. Does this mean that I may not started bringing towels, but intend to start, or does începe just go with the subjunctive in Romanian, is this just a weird sentence...
I'd say a better way to think of the subjunctive is as a secondary verb following the predicate:
I want to run. - Eu vreau să alerg.
I like dancing. - Îmi place să dansez.
Please shut up! - Te rog să taci!
Let me sing! - Lasă-mă să cînt.
She likes to read. - Îi place să citească.
It started raining. - A început să plouă.
I advised you to leave. - V-am sfătuit să plecați.
As you can see, whenever English uses the infinitive or gerund for the second verb, Romanian uses the subjunctive.
Technically, the infinitive is also correct (not sure if in all cases): Eu încep a aduce prosoapele., but it sounds archaic.
It's a simple matter of Romanian using the subjunctive over the infinitive, I don't think there's any traditional "subjunctive reason" to do it.
For you Spanish and French speakers out there, any time you would use the infinitive you use the subjunctive, even though it doesn't follow the normal rules of those languages.
In Bulgarian (and other Balkan languages, like Serbian and Greek) any second verb put next to a first one is connected to it via a specific particle, just as in Romanian. Furthermore, the second verb is conjugated in the same way as the first one. Bulgarian doesn't have an infinitive at all and all verbs are given in the dictionary in their first persom singular form. I believe the trait of adding the second verb this way in Romanian is a very general one of all Balkan languages. I don't even understand why there should be a category for this. Except the differences in some subjunctive forms compared to the normal ones, there is no substantial reason for it being something so special.