A nice paper to understand the connection between Spectra and Motivic Homotopy Theory is Mark Hovey's Spectra and symmetric spectra in general model categories. To form the classical category of spectra you start with Topological Spaces and an endofunctor $\Sigma$ which you wish to stabilize. The trick is to pass to sequences of spaces $(X_i)$ with structure maps $\Sigma X_i \to X_{i+1}$. To do symmetric spectra you build in group actions as you go. In this paper Hovey discussed how to do this in a more general model category $M$ and with a general endofunctor $G$. With enough hypotheses you can prove things like "the stable equivalences are the stable homotopy isomorphisms"

The work of Morel and Voevodsky was the starting place for motivic homotopy theory. They resolved the Milnor conjecture. Model categories came into the picture because they wanted a model structure on the category of schemes. This doesn't quite work, but if you fatten up the category by formally throwing in the missing colimits then it does work. The other trick is to use the Nisnevich site. The punchline is that you now have a category where both topological objects and algebraic objects live. We've done this in a way such that the topological spheres (well, technically simplicial) and the affine line $\mathbb{A}^1$ are spheres, i.e. monoidal units. If you want more details on this work I can add some but I'm going to assume you can go read about it on your own and I'm going to focus on where spectra come in instead.

In the introduction to the paper above, Hovey discusses the application to Morel-Voevodsky. There are several applications. For one, if you want to do spectra in the motivic setting you have to change what you mean by suspension because now there are different types of spheres, as mentioned above. So Hovey's work lets you do this, i.e. pass from unstable $\mathbb{A}^1$-homotopy theory to stable $\mathbb{A}^1$-homotopy theory. Jardine had proven stable equivalences are stable homotopy isomorphisms in the stable $\mathbb{A}^1$-homotopy theory using the Nisnevitch descent theorem. With Hovey's machine (and passage to a nicer model category which is equivalent to the Morel-Voevodsky one) gets this result in a cleaner way without relying on that theorem.

Motivic homotopy theory has become a thriving subject. Jardine has written about motivic symmetric spectra, Po Hu has written about motivic S-modules, and Ostvaer has written about motivic functors (which might be to your liking if you like the Diagram Spectra paper). Googling will give you these papers. I recommend checking out the presentations on Kyle Ormsby's website for an accessible introduction to the field.