Corsage : v. tr. Giving body, consistency to… (Corsage du wine, adding to italcohol).
Increase a note: charge or increase an invoice.
Complicate a problem: make it more difficult to solve.
It thickens or the case thickens : it / gets complicated. Things are getting serious (See below).
Enhance the action of a play, the plot of a drama: reinforce it, increase its interest.
The case, the story thickens: it becomes more complicated, becomes more important, more interesting.
Spice up the sauce: revive a story.
The expression "it's getting tough!" “: it gets complicated! – it becomes more interesting, more intense, more pungent!