The scale tips because you are pushing down on it with your finger. It isn't as obvious that you are pushing on it as it would be if you touched the scale itself. You are pushing on the water, and the water pushes harder on the scale as a result.
The first thing you need to understand is that it doesn't matter how massive your finger is. The size and shape matter. We can simplify a bit if you use a cylinder instead. We will choose a weightless cylinder so that the force of gravity on the cylinder does not confuse the issue. We will also suppose the glass has no mass.
Such a cylinder would float on the surface of the water, not penetrating the surface at all. It would not change the scale. At this time, the water is motionless. The total force on the water is $0$. The two forces on it are gravity and the upward reaction force from the scale holding it in place. These are equal and opposite.
You can make the cylinder penetrate the surface by pushing it down. The water adjusts by rising up the glass a bit, and then stays still. There is an equal and opposite force from the water pushing up on the cylinder.
Now there are three forces on the water, but the water is still motionless. The total force is still $0$. The two downward forces are the force from the cylinder and gravity. The upward force is the reaction from the scale. The reaction force must be bigger now to balance both forces.
From this you can see that it must work out that the downward force from the cylinder must get transmitted to the scale. But how does that happen? To see that, we will look at just the glass of water, and divide it into horizontal layers.
Let us start with one thin layer of water in the glass. The weight on the scale is the weight of the layer. The bottom of the layer presses downward with that much force.
Let us add a second layer. One way to think about it is that twice the water has twice the weight, and so it presses on the scale with twice the force.
Another way is that the top layer presses on the bottom layer with the weight of one layer. The bottom layer presses on the scale with its own weight, but also transmits the weight of the top layer to the scale. So a liquid can transmit a force, even though it is fluid.
You can continue this way with $3$, $4$, or however many layers you like. In each case, the force on the top of a layer is the weight of all the layers above it. The layer adds its own weight to this and presses on the next layer down. The force is proportional to how far below the surface the layer is.
Let's do it again, a little differently. Pour in the first layer and hold the cylinder so the bottom rests on the surface. Nothing new here. The force on the scale is the weight of one layer.
Now pour in enough water so the water rises to a depth of two layers. So what is the difference between this and two full layers? There is a missing plug of water where there is now cylinder. And the force from the cylinder is just enough to keep the water motionless. it is just enough to keep the water from pushing the cylinder upward out of the water. That is just what the weight of the missing plug of water does.
This is key. The downward force from the cylinder must be the same as the downward weight of the missing plug of water. Both are just enough to keep the layer beneath the cylinder motionless. If you push harder on the cylinder, it will sink deeper where the upward force is bigger and will balance it. If you press less, the water will force it upward to where the upward force is smaller and will balance.
Water pushes on the sides of the cylinder as well. The cylinder must push outward enough to resist, or it will be crushed. Again, these forces are the same as those from the missing plug of water, which also must hold back the water around it. These forces on the sides are horizontal, and do not change the downward force on the scale.
So that is what is going on macroscopically, without considering that water is made of molecules. But it doesn't really change anything when you think of molecules.
If molecules were motionless, like little grains of sand, the same explanation would work.
Molecules are really bouncing all around. All the bounces at the top of a layer kick hard enough to hold back all the layers above it. The layer below must kick a little harder to hold back those layers and one more.
There are differences between a gas, liquid, and solid. In a gas, the molecules will travel until they hit something.
In a liquid, they stick together but flow. They vibrate and don't travel away from their neighbors. But they and their neighbors will flow close enough to be in contact with the cylinder.
In a solid, they just vibrate in place. They kick whatever they are in contact with.
A higher temperature means more violent kicks. In a gas, this means molecules kick harder. The forces go up. In a sealed container, raising the pressure by raising the temperature does not change the weight. The gas kicks just as much harder on the ceiling as the floor.
In a liquid or solid, the forces do not go up with temperature. Atoms and molecules stick to each other. If a higher temperature means they vibrate harder, their neighbors hold them back harder.