Ok so it was a bit difficult to find any info regarding this exact structure but I was able to find 3 sentences in the COCA Corpus and based on their whole context I came up with my interpretation of "what play to the base" might mean. Another very useful source was the second website which states near the end that base is not at all related to baseball but to "base" of support either political party has, for example, the Democrats' traditional base is the labor unions.

1- It all seems to me part and parcel of an argument going on around the country among Republicans: can we " play to the base " and succeed, or should we tweak a few things, rethink matters of policy and emphasis and tone?
2- So does that suggest that you don't skew towards the middle, trying to win over those undecided independents but rather play to the base and motivate your supporters to get out there