Will spectral analysis help me understand digital signal processing better?

I am learning Fourier transforms, Z transforms etc. in Digital Signal Processing and I can work easily with integrals.

However, I don't understand how a Fourier transform converts time domain signals to frequency domain signals or why it is invertible. I also have a very basic understanding of linear algebra, not suitable for fourier analysis.

I looked up books on fourier analysis but they don't explain anything I don't already know.

So should the next logical step be to learn spectral analysis? Also, which books would you recommend for linear algebra? I already know matrices, eigenvectors, but not Hilbert spaces.

I would recommend Steven B. Damelin and Willard Miller Jr. "Mathematics of Signal Processing" book. You will find there theory (quite neatly and readable for non mathematicians) and various applications which relate to your question.

It really depends on the level of detail you want to understand Fourier transforms. Fourier transforms and similar integral transforms are very rich mathematical structures that are extremely nuanced. A proper understanding comes from intimate knowledge of measure theory, functional analysis and distribution theory. These are all fairly advanced topics and I suspect not what you had in mind. To understand Fourier transforms at a modest introductory level, texts such as A First Course in Wavelets with Fourier Analysis is sufficient. It outlines the general idea behind Fourier transforms and at the same time highlights some of the difficulties. The text doesn't focus on the nitty gritty details that functional analysts are interested in and handwaves a bit but it is a good text for getting your feet wet in the material. The text covers Fourier series and motivates Fourier transforms from there and then considers wavelet theory (which is pretty nice in and of itself). I think this motivation from Fourier series to Fourier transforms is partly at the center of why you are confused about Fourier transforms and how they extract frequency information. I'll briefly explain it here for the sake of being self-contained somewhat. As a warning: I am going to completely ignore issues of convergence and such but it can all be made rigorous of course.

In regular Fourier series we have that for a suitable function $f$ defined on $[0,2\pi]$ its Fourier series is

$$f(x) = \sum_{n=-\infty}^{\infty} c_ne^{inx},$$

where $c_m = \frac{1}{2\pi}\int_0^{2\pi}f(x)e^{imx}dx$. Note that in the Fourier series expansion, we say that $e^{inx}$ corresponds to the frequency $n$ and the amplitude of that frequency is given by $c_n$ (or more appropriately $|c_n|$).

Now what happens when we begin to extend the interval of definition for the Fourier series to include the whole real line? Enter the Fourier transform. The authors in the text mentioned above go through this procedure and while it is handwaved a bit, it is clear and succinct.

In the case of the Fourier transform we have for a suitable function $f$ defined on the whole real line

$$f(x) = \int_{-\infty}^{\infty}\hat{f}(y)\frac{1}{\sqrt{2\pi}}e^{-ixy}dy,$$

where $\hat{f}(y) = \int_{-\infty}^{\infty}f(x)\frac{1}{\sqrt{2\pi}}e^{ixy}dx$. Note how very similar this is to the Fourier series expressions except that in place of sums we have integrals over unbounded domains. In the case of Fourier series, we called $|c_n|$ the amplitude of the frequency $n$ and we do the same in the case of the Fourier transform, but in the Fourier transform case $|\hat{f}(y)|$ takes the place of $|c_n|$. Now it might seem mysterious that $\hat{f}$ depends on the frequency $y$ but this is the case even with the Fourier series. In general, $c_n$ is a function of the frequency $n$ (hence we index it by $n$). So this is how the Fourier transform allows us to extract frequency information. I hope this answers your question to a reasonable extent.

To add on: the Fourier inversion theorem (as you reference) is actually a very deep result and required a lot of analysis to be developed just to establish it on certain kinds of functions. So to answer the "why" portion is difficult at a rigorous level but at a handwavy level, you can somewhat see how the idea would arise from the Fourier series (where the "inverse" Fourier operation is the Fourier series and the "Fourier transform" in that case is $\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{inx}$).

Disclaimer: If there are any issues with my LaTeX formatting, please feel free to edit my post. MathJax is not rendering for me anymore (I suspect it's a firewall problem or something similar).

If you are more focused on Discrete Fourier Transform, I think this book may be good enough for a reference purposes: http://books.google.ru/books/about/An_Introduction_to_Wavelets_Through_Line.html?id=IlRdY9nUTZgC .
The first part of book devoted do DFT/FFT and (as it is said in the title) it covers them from a linear algebra standpoint.

Spectral analysis is a general term that refers to different fields. If you are interested in using FTs to study time signals, then I can help some.

First of all, this question you have about the "why" of converting time measurements. Think of the surface of a pond. If you drop a pebble in the water, the ripples spread as concentric waves. Drop another pebble nearby and you see two sets of ripples. If the surface of the pond is calm and the ripples are small in relation to the size and depth of the pond, the ripples move smoothly through one another without interacting. The wave amplitudes add linearly to one another. Linear superposition of waves is an important concept. Fourier analysis allows one to describe a complicated function as a linear superposition of waves. Take a note played on a piano. It consists of a simple spectrum of the fundamental, the lowest note, and the weaker overtones that occur at frequencies where the piano string resonates. These frequencies are few because the string is constrained by initial conditions and boundary conditions. The wave description of reality is an alternative to the particle description. Think of billiard balls colliding and interacting. Just as it would be difficult to describe ripples on a pond in terms of colliding particles, it would be difficult to describe the interactions of billiard balls by superimposing waves. Each description has its uses. With the development of EM theory by James Clerk Maxwell, physicists were convinced for a long time that the wave description was the complete answer. The discovery of gamma rays at beginning of the 20th century changed all that, and led to the particle wave duality: both the particle model and the wave model must be used to describe subatomic phenomena.

I would recommend E. Brigham's "The Fast Fourier Transform" for getting down to applications, and also "Numerical Recipes" by W. Press, et al.