Basic confusion about how transistors work
The sentence is talking about the voltage between two points.
It says the voltage at the base must be 0.7V higher than the voltage at the emitter. This is comparing the voltage between two points - the base and the emitter. You can measure this voltage by putting one multimeter probe at the base, and the other at the emitter, and seeing if the multimeter reads higher than 0.7V.
You could also measure the same voltage by putting one multimeter probe on the negative terminal of the battery, and the other one on the base, reading the display, then putting the second probe on the emitter, reading the display again, and then subtracting the emitter reading from the base reading. Or you could use the positive terminal. Or any other good "fixed-voltage" point in your circuit (an audio signal would not work because it would change in between the readings).
It gets rather tedious when describing a circuit if you have to keep saying what two points you are measuring between. It makes life easier to pick a point in the circuit and call it "ground". If the circuit is actually connected to the Earth, then there is a strong convention that the earthed point is ground. Otherwise, it's up to you. The negative terminal of the battery is a popular choice.
By definition, "ground" is at 0V. If you measure the voltage between ground and any other part of the circuit, then you find the voltage "at" that point.
The description of how a transistor works is valid whichever point you pick as ground.
Edit: You could pick the positive terminal of a battery as ground. But that would mean that all the other parts of the circuit would be at a negative voltage. This is valid, if a bit odd. But it's quite common to see dual-rail circuits with a ground, a positive supply and a negative supply.
" Voltage is measured across two points, right? So what is the voltage 'at the base' and 'at the emitter'?"
It's measured from any point you choose. For a grounded-emitter circuit (such as is often used when switching), the emitter is at 0 volts (it's grounded, right?), and when the base voltage increases the current through the base-emitter junction will increase exponentially. For silicon, once the voltage reaches about 0.7 volts there will be sufficient current to get useful amplification.
If you want to pick a different reference point, you might wind up with a voltage from emitter to reference of, let's say, 10 volts. Then, when the base to reference voltage reaches about 10.7 volts the transistor will operate properly.
It's not a matter of the absolute voltage at the emitter, it's the difference in voltages between base and emitter (and the current which this implies) which is important.
It might be simpler to think of the situation as taking a voltmeter and putting one probe on the emitter and the other on the base. That voltage is the one to pay attention to.