Logistic regression provides the odds ratio for each predictor, quantifying how a one-unit increase in the predictor variable impacts the odds of the outcome, assuming other variables are held constant. Specifically, the odds ratio measures the likelihood of an event occurring versus not occurring, given the predictor. For example, an odds ratio of 2.0 means that for each unit increase in the predictor variable, the odds of the outcome happening double.
It depends on context and what I'm trying to do. ORs strip out the base rates so can be useful for comparing things when the base rate is arbitrary (often for more theoretical applications than applied ones). If the base rates are important (which almost always for applied work but not exclusively) then I rarely interpret the OR scale (at least not without also looking at the predicted probabilities). So more often than not I focus not on OR but predicted probabilities (or near equivalents). I'd also nearly always do that graphically unless just dealing with a small set of categorical predictors.
I would like to add to Thom's excellent answer that you could use the plogis() function in R, to get the probabilities, but maybe even more useful, make predicts for several value combinations of your predictors, for example with the "marginaleffects" package. Then it is quite easy to plot this predicted probabilities. Since the regression "lines" are not linear, it is very important to visualize the effects to prevent misinterpretation of the results (in my opinion, non-linear effects are hard to grasp, even if you are experienced, the human mind seems not to be well equiped for this, or is it just me? ;-)
If you want functional relationships that are linear in your model to look linear on your graph, you need to plot predicted log-odds rather than predicted probabilities. Alternatively, you can plot predicted odds and odds ratios on a log scale.
https://andrewpwheeler.com/2013/10/26/odds-ratios-need-to-be-graphed-on-log-scales/