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What do the dots on the tempo's equal sign mean in this picture?

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This means "approximately equal to". I found this with a quick Google search. Here is an example of a webpage confirming the meaning of this symbol. I must confess, I prefer to use "c.", the abbreviation for circa, in metronome markings. Here's an example:

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I've also seen the "wiggly" equal sign used in metronome marks. It's the top one at this webpage (which shows quite a few different symbols for "approximately equal to").

I must confess I hadn't seen the marking in your question, though, so I've learnt something today...

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  • How does this affect the way the music is played?
    – Seafrid
    Commented Mar 7, 2015 at 22:03
  • Just means the tempo marking is approximate; you can play a little faster or slower than this tempo. I would argue it's a bit unnecessary really, as it is rarely that important to play exactly at a specific tempo. Unless, of course, you are playing along with an existing recording, or with a backing track or live electronics, for instance. In which case it should be pretty obvious that you need to play at an exact tempo. Or if playing film music, I guess...
    – Bob Broadley
    Commented Mar 7, 2015 at 22:09
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    Play it at roughly that tempo, possibly a bit faster, possibly a bit slower. Your tempo may (and probably should) vary a bit to take into account the resonance of the room and other factors. This approximation symbol, btw, is commonly used in the Far East.
    – user16935
    Commented Mar 7, 2015 at 22:10

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