Hi @Martin, excellent question. I'll try to provide you off the top of my head with some of the reasons that I personally find the most compelling. I'll try to give you some everyday examples, basically trying to speak your language.
In the end, it all comes down to the fact that evolution makes sense of a lot of stuff we find in God's Creation. God is not a deceiver so we expect His Creation to be in accordance with His Word. Let's see in what way evolution makes sense.
1. All of life looks like a family in terms of anatomy and genetics.
I think this one is often underappreciated because we're being confronted with it every day everywhere. It's so normal that most of us are not fully aware of it. Just taking a look at my own dog, I see four paws, two eyes, one nose, spinal column, one head, similar brain in many ways, even his emotions show correspondence with my own. In his world, punishment is negative and reward is positive. He's hungry like me, pees like me and poops like me. Don't get me wrong, humans are made in God's image and animals aren't. But besides that, the correspondences are so overwhelming that most people are used to taking them for granted. I'm a graduate student in brain research and the correspondences in the functioning of the brain are amazing across species. When you look into the DNA, you find even more of such correspondences (ones that follow the tree of life! ). I'm not an expert on genetics, but Dennis Venema has written a nice series on the evidence of "old genes" in the human genome.
The bottom line of this point is that if God did not want us to come to conclusion that we're all part of the family of life, He could have just made the similarities less extreme. God is not a deceiver.
2. We see a progression of species in the geological record.
This one is less in my street, but the way the species are ordered in the earth is just amazing. We see the most ancient species in the lower layers. As we progress from the deepest layers to the surface, we can basically see the complexity of life arising gradually. Here's a small picture of the distribution in time, which corresponds with what we find in the earth layers:
The bottom line is, if God really didn't want us to come to the conclusion that life arose gradually on Earth, He could have just avoided placing all lifeforms in order of increasing complexity in the geological column.
3. The universe and Earth are unmistakably ancient.
I'm also a graduate student of astrophysics and am very amazed at the ginormous size of God's Creation. Even for distant light to arrive at our Earth it needs billions of years and yet we're observing that all the time. In my studies, we explored in detail how the current picture we have explains the elements that we have on Earth (metal, gold, oxygen, et cetera). Current models even produce predictions of the concentrations of certain elements that were formed in the Big Bang (especially helium) and guess what... it works! Perfect models don't exist, but the current cosmology we have has proven to be astonishingly correct on many different specific predictions. The current Big Bang cosmology agrees with an interesting scriptural truth: Creation as we know it had a definite beginning. Before, scientists like Einstein preferred to belief that the universe was eternally existing and stable. However, later the evidence for a beginning became so strong that most people accepted the Big Bang model.
4. There are currently no other models that can answer to the huge mountain of findings that has been piling up (and still growing).
Especially the website Naturalis Historia of Joel Duff has been very useful for me in understanding the blatant inconsistencies of the young earth paradigm. He's a biology professor but also explores archaeological issues. For example, we find gazillions of stone tools in Africa, which can't be placed in a short timeframe of 10,000 years. Even if every single human being in Africa would have produced one million stone tools during his or her lifetime you would fall short of the actual number: 15-150 trillion stonetools (!!!).
5. Important one: I don't find evolutionary theory (as a scientific framework) to be in contradiction with the Christian faith and the teachings of the Bible.
We have to look at what the Bible aims to teach, not at the ancient cosmology that the writers were familiar with. One interesting observation is that even the writer of Genesis was comfortable with the idea that God gave the Earth the ability to bring forth life. In Genesis 1:24 it is written:
And God said, “Let the earth bring forth living creatures according to their kinds..."
The final bottom line is... If God's Creation speaks to us, we do well to respect what it's telling us. Since it isn't contradicting what God revealed through the Scriptures, we can rest assured that there are no risks involved if we remain critical of the science.