The failure rate of the exponential distribution is a constant, $\lambda$, as the exponential distribution is memoryless.
So say we have that $\lambda = \frac{1}{10}$. What is that telling us?
The item has a one in ten chance of failing in the next instant? But as we can make the next instant as small as we want, even if it doesn't fail in this 'first' instance, we will immediately have a $1$ in ten chance of failing in the second instance, which is infinitely close to the first instant, etc.
This seems to say that the value of $\lambda$ is irrelevant and the item will always fail infinitely quickly. So I don't understand the failure rate function. What does a failure rate of $\lambda = \frac{1}{10}$ signifiy?
Failure rate, or hazard rate, or force of mortality of a device/object/life at some time $t$ is best interpreted as the instantaneous exposure to failure/death given survival to time $t$. Formally, $$h(t) = \frac{f(t)}{S(t)} = \lim_{\Delta t \to 0} \frac{\Pr[t < T \le t + \Delta t \mid T > t]}{\Delta t},$$ where $T$ is the future lifetime random variable. This can also be regarded as a likelihood of failure at time $t$ given that survival up to time $t$ has occurred.
Because the numerator is a probability density, the hazard rate is not itself a probability: therefore, it is incorrect to interpret it as such. Indeed, it is easy to see that $h(t)$ need not be less than $1$.
A constant hazard rate simply (but loosely) means that there is constant exposure to failure at any time: for small time increments, the probability of failure in a given increment is proportional to the length of that increment.