I have computed the IV estimate and the standard error of the IV estimate. Why do I care about its standard error?
By "IV", I will assume you mean independent variable, but it is worth spelling these things out as they can stand for various things in various contexts; for example "IV" can stand for instrumental variable. Furthermore, by "computed the IV estimate", I assume you mean that you estimated the coefficient in a standard (ordinary least squares) simple regression model, or something comparable (n.b., many basic analyses, such as $t$-tests and ANOVAs are special cases of regression models, see here: How are regression, the t-test, and the ANOVA all versions of the general linear model).
So, why do you need the standard error (SE) of your estimate? Well, you don't necessarily need the standard error of your estimate. The estimate gives you the slope of the best fitting line in a regression model, or perhaps the mean of a category in simpler situation. That may be all you need in some cases, and if so, it is perfectly reasonable to stop there.
However, you may want to know the standard error as well. The standard error provides some information about the uncertainty of your estimate. That is, how much on average will the estimate bounce around from one study to the next if you were to re-run your study from scratch, gathering the same amount and type of data in the same way.
Knowing the SE makes statistical inference possible. That is, you can test if your estimate is sufficiently far from some reference (null) value, often 0, that you might conclude it is not reasonable to believe that the data actually came from a population where the reference value was the true value.
It is worth bearing in mind that the SE relies much more strongly on assumptions than the estimate itself does. The estimate, if it is a slope for a regression line, only relies on the assumption that the model's functional form is correctly specified (e.g., the true data generating process is a straight line). If the estimate is a simple group mean, it doesn't even assume that. On the other hand, the standard error also relies on the assumptions that the data are independent, that the variance is constant, and that the conditional distribution of the data on the model is truly normal (or that it is close enough to normal that the Central Limit Theorem will allow sufficient convergence of the sampling distribution to normality given the amount of data you have). These assumptions are more restrictive, and need to be checked in order to ensure the validity of your SE. (Note that there do exist 'robust' SE's for various situations in which the above assumptions do not hold, such as the sandwich estimator for the SE.)
To learn more about standard errors, it may help you to read my answers here: How to interpret coefficient standard errors in linear regression?, and here: How does the standard error work?