Why is it that Complex Numbers are algebraically closed?
Of course there are many proofs, and perhaps some others will post the most attractive proofs, but I think you are looking for an intuitive explanation that would somehow make the result seem less surprising.
One such explanation, I think, is the simple observation that reals already go a long way towards being algebraically closed---they are a real closed field---since every odd-degree polynomial over $\mathbb{R}$ has a root in $\mathbb{R}$. This follows immediately from the intermediate value theorem, since in the large scale every odd degree polynomial moves from $-\infty$ to $\infty$ or conversely and hence must cross the axis.
Here are 3 facts that I think provide some kind of intuition :
To show that $\mathbb{C}$ is algebraically closed, you only need to show that real polynomials have a root in $\mathbb{C}$ : because of Taylor's formula, a complex polynomial taking real numbers to real numbers is actually real. Now if $P \in \mathbb{C}[X]$ is a complex polynomial, then $P\overline{P}$ is a real polynomial and has the same roots as $P$.
All odd degree real polynomials have a root in $\mathbb{R}$ (because they are continuous and the limits at $-\infty$ and $\infty$ have different signs).
Because of the quadratic formula, solving degree 2 equations only requires taking square roots, which is always possible in $\mathbb{C}$ because of the geometric interpretation and because square roots of positive numbers exist in $\mathbb{R}$ (if you write $z = \rho e^{i \theta}$, then a square root of $z$ is given by $\sqrt{\rho} \ e^{i \theta /2}$).
From these 3 facts (notice that they use some analysis) and some clever algebraic manipulations (which you can find on Wikipedia in the section "Algebraic proofs"), you can deduce that $\mathbb{C}$ is algebraically closed. This is in my opinion the closest you can get to an intuition.
I think your question can be boiled down to understanding why the fundamental theorem of algebra is true. As has already been pointed out above there are many different ways to prove this and you should try to understand a few of them. However your question is not really about the mechanics of those different proofs but rather you are asking 'Why are the complex numbers enough..?'
It's a good question. The truth is that each of those different proofs is giving an argument about why they are enough -- from a slightly different perspective. Depending on your background you may find one more intuitive than another. I have two suggestions on how to get a better feel for the FTOA:
Perhaps what you are looking for in an intuitive answer is something visual. The nice thing here is that you can make awesome colorized pics which reveal the structure of complex valued functions. Check out this unpublished paper by Daniel J. Velleman at Amherst. It's a great visual walk through of a few approaches to the FTOA. It's really a very nice read and the plots bring it all together in a way that many people feel is intuitive. If you're good with coding then you can leverage something like SAGE or mathematica to make some plots of your own and understand the reasoning of the FTOA with your own examples too!
If that doesn't lock it in then take a stab at reading through Fine and Rosenberger's book. You can find it in most university libraries with a strong math department. They'll walk you through the FTOA from three different perspectives; algebra, complex analysis, and topology. It's a longer approach perhaps but I suspect it will bring a lot of mathematical loose ends together for you.
Best of luck and success in your studies! You've asked a great question and that's where it all starts.