I am looking for a simple explanation of the implication of having homothetic/nonhomethetic preferences in relation to consumers' preferences when consuming goods.
From a mathematical point of view, if the function $f(x,y)$ is homogeneous (of any degree), and $g()$ is a function whose first derivative is everywhere non-zero, then the function
$$H(x, y) = g[f(x,y)]$$
is homothetic. In economics, we usually impose something more restrictive, namely that $g' >0$. But this makes a homothetic function a monotonic transformation of a homogeneous function. Now, homogeneous functions are a strict subset of homothetic functions: not all homothetic functions are homogeneous.
Therefore, not all monotonic transformations preserve the homogeneity property of a utility function. The simplest example is Cobb-Douglas utility. It is homogeneous of degree one. In an ordinal utility framework, we are ok with monotonic transformations, so we can consider the natural logarithm of it. Fine, but the natural logarithm will not preserve homogeneity. Nevertheless it will be homothetic.
The fundamental property of a homothetic function is that its expansion path is linear (this is a property also of homogeneous functions, and thankfully it proves to be a property of the more general class of homothetic functions).
In consumption theory, this means that, keeping the prices or the price ratio constant, if we vary the income of the consumer, in the $(x,y)$ plane the tangency point of the income constraint with the highest feasible indifference curve will always reflect a fixed ratio $x/y$. This in turn implies that expenditures for each good grow all at the same rate as income, and so expenditure shares remain constant for the whole income range (always for a given price ratio). While this may sound restrictive, in fact, it has been shown that homothetic preferences do not impose any special restrictions on aggregate demand (essentially due to the endowment vector being arbitrary and independent of preferences, which messes, or frees, things up).
Well, if the preferences are homothetic, they will have a bunch of cool features. Formally, if they are homothetic, then you can affirm that: $$U(\alpha x, \alpha y) = \alpha U(x,y), \forall \alpha > 0$$
Something cool about this kind of functions is that they are, as you can see from the definition above, homogeneous functions of degree 1. So, you have the "right" of one normalization of the arguments. That's why when solving for the indirect utility function, you can always set one of the prices as the numeraire and set its price as 1.
Another interesting feature is that the goods demanded by consumers in a competitive model will depend only on the price ratio, not income.