Understanding the Ground Symbol
For the sake of this circuit, the ground symbol is not strictly necessary for understanding the voltage divider. It simply alludes to the fact that this is the point where the (-) node on the battery would be attached to an earth ground of some type when actually put in an enclosure.
This takes a minute to understand for the first time.
When you have any voltage, you are really talking about the difference in voltage between two nodes. If no second node is specified, you are supposed to imply by convention that they are referencing from ground (often referred to as
common in DC circuits).
It's like asking "What's the elevation of this computer?" You might mean the elevation relative to sea-level or you might mean relative to the floor, but you could also take elevation from, say, your kneecaps or the ceiling or anything else you can think of.
So what a battery actually does is it creates a voltage difference between the negative terminal and the positive terminal of the battery.
So then, what is the point of having an actual ground symbol tied to the (-) terminal of the battery? There are a few reasons. First, every time moving current is present, there will be a magnetic field induced that curls counter-clockwise around the core of the wire (right hand rule). Furthermore, in any environment there could be random magnetic fields coming from the lights, from radio waves, from the sun, and so on. You could also gain static electricity which is influenced by the humidity, the materials of the environment, etc. Because all this stray magnetic radiation could be present near your circuit, and any pieces of metal can act like an antenna which transform magnetic field into electrical current, your circuit could get unwanted and unplanned electrical current, which could interfere with what the circuit is designed to accomplish and in some cases it can be unsafe.
So the solution is to designate a terminal to be ground and then connect the (-) terminal of your voltage supply to it, along with the chassis of your enclosure, or if you a connecting to a building's electrical outlet, a connection to the earth ground, which is actually a physical pole buried literally in the ground, as in the dirt.
So essentially, if you left the ground symbol off of your voltage divider circuit, you could analyze it exactly the same way, and you wouldn't have any problems until you actually built the device and you might have noise.
Hopefully this explains the ground. Now to actually analyze the circuit.
Analyzing the Circuit
You have four terminals shown, so you can get voltages (referenced to ground) on all four. Let's number these 1-4 from top to bottom.
1 should be straight-forward. It's 15V: it's connected directly to the (+) terminal of the battery.
4 should be straight-forward as well for the same reason. It's 0V. What's the difference between ground and ground? Zero.
Now the key for understanding 2 and 3 is that you assume that current cannot actually flow out of the terminals. For one, it's not hooked up to anything. For two, generally in circuits that deal with output voltages from one device they are going to be hooked up to devices with inputs that have "infinite" impedance (i.e. resistance in this context). I put infinite in quotes because it's technically not infinite, but for all intents and purposes you can assume that the resistance is so large that the amount of current that can actually escape might as well be considered zero.
This means that current has exactly one path to ground, and it is through the resistors. This means that the current going through these resistors must be the same (see Kirchoff's Current Law).
So you can get the total resistance of the circuit by adding
R1 + R2 + R3 = Rtotal. You can solve for the current using Ohm's law. V = IR, so
15V/Rtotal = I
Now you can go back and notice that I * R1 produces a voltage drop. So
15V - I*R1 = Voltage of Terminal 2.
Now you can notice that I * R2 produces a voltage drop as well. So
Voltage of terminal 2 - I*R2 = Voltage of terminal 3.
Now you know voltages of all nodes.