The null hypothesis in each case is that the variance due to each of
the factors is zero, or neglible, compared to the error variance. We test
this by taking a ratio of the variances (mean squares), the F test.
If F is big, then the variance is big compared to the "random" or "unexplained"
variance. The P value (called in this table, the "significance") is a measure
of the probability of getting the observed F value by chance if the Null
Hypothesis were true. If the probability of getting this observed F is
low, then we can, with some confidence reject the null hypothesis and say
the effect is "statistically significant". We choose a level of confidence
ahead of time depending on how willing we are to be wrong in rejecting
the null hypothesis when it is true. We picked 5% for alpha in this
case- so we are willing to accept that type 1 error 5% of the time.
What we see in the ANOVA is that both balls and clubs give significantly large F values. The clubs effect is HUGE, with an F of 937, compared to a critical f, 5% of 4.25. The Probabilitiy of getting this by chance if there were really no effect is about 10 to the -20th power. that is unlikely.
the balls effect gives an F of almost 8, compared to a cutoff critical F,5% of 3. The Pvalue is .0008, much less than 5%, and so we reject the null hypothesis. ANOVA has allowed us to see the ball effect by controlling for the club effect.
the interaction is also significant F=7 > Fcrit,5%=3, P=0.001, meaning that the ball effect changes in magnitude depending on which club you use, and that this change is statistically significant, giving us an observed effect that is unlikely (p=0.001) to happen by chance --i.e.if the null hypothesis were true and the effect wasn't real.
Within Mean square is simply the unexplained, or error, or random variance. It's the part of the world we call "random" because we don't understand it yet.