Is multicollinearity the issue here? How is that I get a low $R^2$ if I use $X1$ against $Y$ and $X2$ against $Y$, but a $R^2 =1$ if I use both together? 
How does the combination blow out the $R^2$?
Could you please show me that calculation?
The data:
Y   12.37   12.66   12.00   11.93   11.06   13.03   13.13   11.44   12.86      10.84    11.20   11.56   10.83   12.63   12.46

X1  2.23    2.57    3.87    3.10    3.39    2.83    3.02    2.14    3.04    3.26    3.39    2.35    2.76    3.90    3.16

X2  9.66    8.94    4.40    6.64    4.91    8.52    8.04    9.05    7.71    5.11    5.05    8.51    6.59    4.90    6.96

 A: In multiple regression, you are trying to model observations, $y$ that are supposed to be noisy versions of a prediction, $X\beta$. That is, we should have:
$$y=X\beta+\epsilon$$
In your case, $\epsilon$ is almost zero. That is, both explanatory variables almost perfectly explain your observations. You have:
$$\beta = (-4.515413640943276 \text{ (constant offset)}, \\ 3.097007886142981, \\ 1.031859030944453)$$
This gives you a nearly perfect fit.

Regarding multicolinearity: By definition, multicolinearity occurs when your predictor variables, $X1$ and $X2$, are linearly related.
Now, in your case, $X1$ and $X2$ do, in fact, have a high correlation coefficient of about -$0.9$ (I had missed this in a previous edit).

However, this is not the cause of the $R2$ jump that you observe by including both $X1$ and $X2$.
For, let multicolinearity have been present. Then, we could write for a small error $\eta$:
$$X2 = a + bX1 + \eta$$
In that case, we would be able to write, for each data point $i$:
$$y^{(i)} = \beta_0 + \beta_1 X1^{(i)} + \beta_2 (a + bX1^{(i)} + \eta^{(i)}) + \epsilon^{(i)}$$
$$\Rightarrow y^{(i)} = (\beta_0 + a \beta_2) + X1^{(i)} (\beta_1 + b \beta_2) + \epsilon^{(i)} + \beta_2 \eta^{(i)}$$
Now that we know that $\epsilon^{(i)}$ are nearly zero, then, unless $
\beta_2$ were large, had multicolinearity been an issue we would have seen $R2 = 1$ for even the $Y-X1$ regression. Instead, we get $R2 \approx 0$. This means that while $X1$ and $X2$ together are good predictors, they aren't linearly related enough to be good predictors individually.
Having said that, multicolinearity might play a role in other aspects of your problem, given the $X1-X2$ correlation coefficient of $-0.9$.
