It all boils down to theory vs. practice.
The true value of a parameter is always a theoretical quantity. Thus, you can never determine the true $\theta$.
The idea behind this is, that there is some kind of process, which generates the data. This process has some parameter. If you knew that parameter and the process you could generate data, which would be indistinguishable from the real process. Other parameters always generate data, which can eventually be distinguished from the real process.
The only way to "know" the true $\theta$ is if you define the process mathematically. For example, if you define a fair coin, then the true probability of showing heads is $0.5$, but not because you determine it by an experiment, but because you defined it to be.
A consistent estimator will tend (probabilistically!) towards the true value, which is captured by the definition you provided. As you collect more data, the probability that the consistent estimator differs by a certain amount $\epsilon$ from the true value goes towards zero (NOTE: it is a common misconception that probability of zero means something never happens, but that is not necessarily true. It only means, it practically never happens).
So if you want to "determine" a true value just based on the data alone, a consistent estimator is your best method. You can never be sure, that you really got the right value (which is why any estimator should come with confidence intervals, etc).
Now, how do you know an estimator is a consistent estimator if you can't know the true value in practice? If some estimator is consistent is not a practical observation, but a theoretical property. Thus, you prove (theory!) that something is a consistent estimator.
Take for example the expected value and variance (true parameters) of a normal distributed variable (the process). The mean of the samples is a consistent estimator of the expected value. So you define $X \sim Normal(\mu,\sigma^2)$ (i.e. $\theta = (\mu,\sigma)$), and then you prove (!) that the mean will tend probabilistically towards $\mu$ for any $\mu$ and $\sigma$. Then you know, that the method you have just proven can be applied in practice to estimate the true $\mu$.