Intuition behind sphericity assumption
One of the assumptions of common, non repeated measures, ANOVA is equal variance in all groups.
(We can understand it because equal variance, also known as homoscedasticity, is needed for the OLS estimator in linear regression to be BLUE and for the corresponding t-tests to be valid, see Gauss–Markov theorem. And ANOVA can be implemented as linear regression.)
So let's try to reduce the RM-ANOVA case to the non-RM case. For simplicity, I will be dealing with one-factor RM-ANOVA (without any between-subject effects) that has $n$ subjects recorded in $k$ RM conditions.
Each subject can have their own subject-specific offset, or intercept. If we subtract values in one group from values in all other groups, we will cancel these intercepts and arrive to the situation when we can use non-RM-ANOVA to test if these $k-1$ group differences are all zero. For this test to be valid, we need an assumption of equal variances of these $k-1$ differences.
Now we can subtract group #2 from all other groups, again arriving at $k-1$ differences that also should have equal variances. For each group out of $k$, the variances of the corresponding $k-1$ differences should be equal. It quickly follows that all $k(k-1)/2$ possible differences should be equal.
Which is precisely the sphericity assumption.
Why shouldn't group variances be equal themselves?
When we think of RM-ANOVA, we usually think of a simple additive mixed-model-style model of the form $$y_{ij}=\mu+\alpha_i + \beta_j + \epsilon_{ij},$$ where $\alpha_i$ are subject effects, $\beta_j$ are condition effects, and $\epsilon\sim\mathcal N(0,\sigma^2)$.
For this model, group differences will follow $\mathcal N(\beta_{j_1} - \beta_{j_2}, 2\sigma^2)$, i.e. will all have the same variance $2\sigma^2$, so sphericity holds. But each group will follow a mixture of $n$ Gaussians with means at $\alpha_i$ and variances $\sigma^2$, which is some complicated distribution with variance $V(\vec \alpha, \sigma^2)$ that is constant across groups.
So in this model, indeed, group variances are the same too. Group covariances are also the same, meaning that this model implies compound symmetry. This is a more stringent condition as compared to sphericity. As my intuitive argument above shows, RM-ANOVA can work fine in the more general situation, when the additive model written above does not hold.
Precise mathematical statement
I am going to add here something from the Huynh & Feldt, 1970, Conditions Under Which Mean Square Ratios in Repeated Measurements Designs Have Exact $F$-Distributions.
What happens when sphericity breaks?
When sphericity does not hold, we can probably expect RM-ANOVA to (i) have inflated size (more type I errors), (ii) have decreased power (more type II errors). One can explore this by simulations, but I am not going to do it here.
It sounds like you're looking to test one basic hypothesis on four similar variables. Setting aside the thought of estimating a latent fitness factor from these four variables, you can approach this with a mixed effects MANOVA. Because you expect fitness in general to change differently for your two groups, and you have four indicators of fitness, you can test your hypothesis of group differences on each of your four dependent variables while controlling for multiple comparisons using MANOVA.
As I understand your problem, you want to include a random effect for individuals measured repeatedly. Your fixed effect is training program. You have four dependent variables. You expect no group differences at the first measurement, but expect group differences at the second, and I'm guessing you expect those differences to be stable until the third measurement. You can test group differences and the group variable's interaction with measurement time in repeated measures ANOVA for each of your four dependent variables, but MANOVA makes the test more conservative by controlling for familywise error rate inflation caused by taking four separate whacks at the hypothesis.
I should warn you that these general linear models are conventionally fitted by ordinary least squares estimation, which may produce biased estimates of standard errors, and therefore biased significance test results if your data don't meet the assumptions...and real data often don't. Also, there's some controversy regarding the utility of controlling for familywise error inflation. If you want to make sure you're choosing the right analysis for your purposes and care to study related issues, @HorstGrünbusch's link to this question is definitely a good one to follow too:
Best Answer
Having several repeated-measures DVs one can apply a univariate approach (also called Repeated Measures sensu stricto or split-plot approach) or multivariate approach (or MANOVA). In univariate approach, RM levels are treated as deviations from one variable, their average level. In multivariate approach, RM levels are treated as covariates of each other. Univariate approach requires sphericity assumption while multivariate approach does not, and because of this it is becoming more popular indeed. However, it spends more df and thus needs larger sample size. Also, univariate approach retains its popularity because it generalizes to Mixed models. When sphericity assumption (and beyond expectation more general compound symmetry assumption too) holds results by both approaches are very similar, as far as I know.