It is because you have 101 points in your signal. Therefore the magnitude of the DFT at the frequency is going to be approximately N/2 where N is the number of points. If your sine wave had an amplitude other than 1, you would see NA/2
To make this exact, let's create your sine wave with 100 points so that the frequency of 10-Hz falls directly in a DFT bin
t = 0:0.01:1-0.01;
x = cos(2*pi*10*t);
xdft = fft(x);
plot(abs(xdft))
Now you see the magnitude at -100 and 100 Hz is 100/2=50
Change the amplitude of the sine wave:
x = 3*cos(2*pi*10*t);
xdft = fft(x);
plot(abs(xdft))
Now the magnitude at -100 and 100 Hz is (3N)/2=150
This dependence on N comes from the fact that the DFT sums all the element-by-element products of your signal and complex exponentials with frequencies of (kFs)/N where Fs is the sampling frequency and N is the length of the signal and k=0,1,2,3, N-1
When your signal matches the frequency of one of those complex exponentials exactly, as is the case above, you get the coherent sum that results in N times the amplitude. Since your input is a cosine remember that results in TWO complex exponentials each with amplitude 1/2 the amplitude of the cosine.
