The key to team improvement is good retrospectives.
When I started running retrospectives, I found they often just degenerated into whingeing sessions that didn't achieve much, other than giving the team an opportunity to vent our frustrations.
Now I go into the retrospective with a clear goal, stated at the beginning of each one:
We're here to find what we, the team, most want to improve, and decide on how we'll do that.
I keep tight time-boxes on the main phases:
- Summarise what happened in the sprint.
- Individually, write good/bad/ugly memories on stickies.
- As a team, group the stickies into themes, and pick the most important one (possibly two, but no more).
- Brainstorm ideas for the improvement activity (i.e. sustain a good thing, or reduce a bad thing).
- Refine the best ideas so that they are concrete actions with a result you can measure (look up SMART criteria if you haven't already).
Remember, as a Scrum Master, your role isn't necessarily to make the improvements yourself; the key trait is to facilitate the team. So the important thing is ensuring that the Retrospective takes place, and kept on track towards its outputs (the improvement activities).
Once you have the improvement activities agreed, then the Scrum Master can be monitoring the agreed metric, and checking that the improvement activity is actually happening. This can involve reminding team members of what they agreed to do in the meeting; I find it helpful to have the list of improvement actions visible in the stand-up area, so that we are all reminded of them every day, and can have a quick check of our progress towards them as well as to our sprint goals.