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Shortly afterwards German politics will turn inwards ahead of a parliamentary election in September, probably re-engaging in earnest with Brexit only in November, once a new coalition has emerged in Berlin.
Does it mean the election will soon happen? Or Germans will care more domestically because of the election?
Yes, it's talking about the Germans focusing more on domestic issues (the inward part of the country) as the parliamentary elections near.
There are two separate parts to the phrase that you're asking about:
Ahead of means "before".
Turn inwards means something like "focus more on internal thoughts and matters" (like when making a decision), instead of worrying about external influencing forces.
In this specific example, the author is saying that Germans will care more about the German parliamentary election, since it's internal, and will only pay attention to Brexit again once the election is over.
