The term 'regression' means : 'deterioration', 'moving back', 'to return to an earlier or previous state'. So when we speak about regression analysis, what does the word 'regression' mean? Which & what sort of 'deterioration' or 'going back' are we analyzing?
The original use by Francis Galton was in a problem where children's heights were being predicted from parents' heights. Tall parents (parents taller than average) tend to have tall children, but on average shorter than themselves, whereas short parents (parents shorter than average) tend to have short children, but on average taller than themselves.
This phenomenon was called regression to the mean, where regression means falling back, rather than deterioration.
The name has stuck for all similar problems, including the great majority that have nothing to do with heredity. But a regression effect can be expected in many testing-retesting problems.
This history is well documented. For example, see this Wikipedia article and its references.
A regression analysis with $k$ predictors has a form like this $y= a + b_1 x_1 + \cdots + b_k x_k$. Thus the regression regresses a dependent variable on its predictors.
Here is a similar question and part of the answer is "Since $y$ is being projected on $X$, that is what I think when I hear that $y$ is "regressed on $X$" - and that is probably why we talk about "regression" analysis. The link provided may make the question to a duplicate.