Early propagation of carries in the BLAKE hash function
This is already covered by Ella's response, but in short the added nonlinearity at the start buys you very little in terms of security, compared to its cost in number of operations. It would be more effective, and make for a simpler design, for example, to trade all these constant xors by an extra round. Since that part of BLAKE was not pulling its own weight, as noted by the cryptanalysis of the various weakened versions of BLAKE, it was removed.

But besides the small effect on nonlinearity, there is another reason for the existence of the constants—to eliminate symmetries on the underlying block cipher. BLAKE, as well as BLAKE2, are modes of operation on top of a $\{0,1\}^{1024}\times\{0,1\}^{1024}\to \{0,1\}^{1024}$-bit (or $\{0,1\}^{512}\times\{0,1\}^{512}\to\{0,1\}^{512}$) block cipher $E(k, m)$, inspired by the ChaCha core. And like the ChaCha core, when the message words $m_i$ are all 0, or more generally all equal, if the inputs $x_i$ are also, say, all equal, the output words $y_i$ will also be all equal. For example,
$$
E\left(
\begin{pmatrix}
m &amp; m &amp; m &amp; m \\
m &amp; m &amp; m &amp; m \\
m &amp; m &amp; m &amp; m \\
m &amp; m &amp; m &amp; m \\
\end{pmatrix},
\begin{pmatrix}
x &amp; x &amp; x &amp; x \\
x &amp; x &amp; x &amp; x \\
x &amp; x &amp; x &amp; x \\
x &amp; x &amp; x &amp; x \\
\end{pmatrix}
\right) =
\begin{pmatrix}
y &amp; y &amp; y &amp; y \\
y &amp; y &amp; y &amp; y \\
y &amp; y &amp; y &amp; y \\
y &amp; y &amp; y &amp; y \\
\end{pmatrix}\,,
$$

for some 32- or 64-bit word $m$, $x$, and $y$. This property, often called a symmetry of the round function, makes the underlying block cipher of BLAKE—without the constants—easily distinguished from an ideal cipher. Likewise with BLAKE2. This is even exploited in the "chosen-IV" attacks of a weak variant of BLAKE2. 

However, in the actual mode of operation of either BLAKE or BLAKE2, these so-called symmetric states are unreachable by design, via the inclusion of the $IV_0$–$IV_7$ constants in the initial state, so this property is not a security liability.