Dave's right (as usual). It's a byproduct of pruning the
lateral branches (AKA suckers). The plant wants/needs more leaves I guess.

I assume you'd just cut off beyond the fruit.

This growth is more of a factor of other stresses, notably excess nitrogen fertilization, occuring with plant growth and I find it very rare in most hybrids. I do however see it more often with a few grape varieties I grow. Regardless of what the cause you don't want to allow branches to originate fron a blossom cluster as that creates a weak, dangling, stem and interfers with fruit development.

To the contrary of what you might hear suckering does cause larger fruit, overall earlier harvest and therefore a higher percentage of high quality fruit. Less disease due to better air circulation is an often overlooked added benefit.

The other problem you mentioned of high blossom numbers may correct itself as some fail to develop or you may just have more small fruit formed if you don't cull a few fruits early in the development stage. With many larger fruiting heirloom varieties I cull double, fused and otherwise severly misshaped blossoms as they seldom produce a marketable quality fruit.
With many heirloom varieties the advice of Dr. Adam Montri, Michigan State, to remove all suckers except to allow 2 stems per plant seems to make more sense.

Dr montri's advice appears to make more sense to me in my GH house. I am a pincher and have found better results with indeterminate plants - having tried it both ways. I allow 2 branches and pinch out all other suckers - acknowledging that there is more than one way to prune tomatoes. Bmoser's post mirrors my conclusion and not cutting these branches would definitely weakin the fruit cluster and the branch. I have never seen this condition on Better Boy indeterminate plants growing right next to the heirloom varieties in the same soil with the same fertilizer (0-52-10) but it is nice to know the condition is common and no big deal cutting them out. Having never grown a heirloom indeterminate variety - I may let one plant keep it's suckers and compare the results?

Last years tomato crop - pinching out all the suckers and pruning the leaves as the the fruit on each tier was harvested. We had tons of tomatoes !!

I agree. pinching suckers is another name for pruning . I do that mostly to the lower branching to have good air movement and discourage moisture based plant deseases.
And also, because I plant them tightly( less than two feet) I have to prun and keep plant size under control.

So to prun or not to prun is gardner's choice and style.