How does time factor into Cox regression or a Cox proportional hazards model?

The hazard function for the Cox proportional hazard model is written as follows.

$$\lambda(t|X_i) = \lambda_0(t)\exp(\beta_1X_{i1} + \cdots + \beta_pX_{ip}) = \lambda_0(t)\exp(X_i \cdot \beta)$$

I would like to know how come the time, $t$, is not really used in prediction (this link states that the hazards ratio does not depend on time). I know that one of the assumption is that the effects of the covariates are constant over time, but the $t$ being used as an input would imply that it has some influence in the prediction.

Furthermore, in this example, there are 2 subplots: one with the cumulative hazard and one with the survival predicted for an individual. What is the y-axis in each of these or how do I interpret these plots? The second plot seems clear to me: it is the probability of survival over time. The first plot does not seem so clear: is it the relative risk ratio over time (meaning at 50 years, the relative risk ratio is 6 times the baseline hazard)?

Any help is appreciated.