Here's a high level breakdown of Dijkstra's algorithm:

You stick all of the vertices in a priority queue where all of the vertices have a priority (distance) of infinity except for the source vertex, which has a distance of zero (the source vertex is zero units of distance away from itself, right?).

Pop the priority queue. The vertex removed is the vertex with the shortest distance from the source. Obviously the first vertex popped from the queue is the source. Well call that popped vertex v.

Look at each of the neighbors of v. All of them will have a distance greater than v's distance (otherwise we would have already popped them from the queue, right?). Let's say v has a distance of 3, and v has 3 neighbors x (dist-from-source: 5), y (dist-from-source: 10) and z (dist-from-source: infinity).

Now we look at each neighbors distance from v. Let's say they are: x - 3, y - 2, z - 4. This means that the path from the source to x that goes through v has a distance of |v| + 3 (3 + 3 = 6), y has a distance of 5 (3 + 2 = 5) and z has a distance of 7 (3 + 4 = 7).

The path to x through v is longer than the current shortest path to x so we ignore it. However the paths to y and z that go through v are shorter than the previous known shortest path so we update them.

You keep doing this until you have gone through all the vertices. At each point, when you pop the min from the priority queue you know you have found the shortest path to that vertex because any possible shorter path must pass through a vertex with a distance less than v's, which means we would have already popped that from the queue.