I might have some of this wrong so someone might have to correct me, but this is how I see it...
What's important to realize is that Chlorine and Fluorine both want to accept an electron to achieve an octet electron configuration. Given that fluorine has its electrons (both core and valence) held closer towards the nucleus, it is denser. According to Coloumbs law, the closer you bring two charges towards one another, the greater the electrostatic force. Since the charges in this case (electrons) are both negative, the electrostatic force of repulsion increases. Therefore, to add another valence electron to an atom that is denser (has its electrons held tighter) will become more difficult as the electrostatic repulsion increases. Chlorine has a similar issue, but its electrons aren't held as tightly, so it is more favorable for it to accept an electron. Keep in mind that the tendency of either fluorine or chlorine to accept an electron is still very high. Both will do it since achieving an octet is thermodynamically favorable. It just turns out that its more favorable for chlorine to do it relative to fluorine.