Agree with others here. Stem plants will develop leaves and roots from nodes along the stem, and depending how you grow them, new shoots. Some species do this in what might appear to be excess, while others minimally. But this also allows you to cut the stem at any point and planting the growing end as a new plant (provided you leave several nodes on the cut stem).
I've heard it mentioned before that plants do this to compensate for not obtaining enough nutrients from the substrate with their primary roots. Whether this is true or not, it doesn't apply to Ludwigia, this plant will grow shoots from every leaf node regardless. You can trim them if you think they're unsightly.