Note I'm not sure how to explain diagram chasing except by giving an example, so I'll go through the proof, and then try to explain how this is diagram chasing.
Setting up terminology:
Let $X$ be a category, and $(T,\mu: T\circ T\to T,\eta : 1_X\to T)$ a monad on $X$.
Let $H : I \to T\newcommand\Alg{\text{-}\mathbf{Alg}}\Alg$ be a diagram of shape $I$.
Let $G:T\Alg \to X$ be the forgetful functor. Let
$(L,\sigma)$ be a limiting cone to $GH$ with $L\in X$ the vertex and $\sigma_i : L\to GH_i$ the morphisms of the cone.
We need to show that unique lift of this cone to $T\Alg$, and this lift will also be a limiting cone. Since the forgetful functor is faithful, this amounts to showing that there is a unique algebra structure on $L$ making $(L,\sigma)$ a cone in $T\Alg$ and then showing that this cone is a limiting cone.
The algebra structure on $L$
An algebra structure on $L$ is a morphism $\alpha_L : TL\to L$ such that $\alpha_L\circ \eta_L = 1_L$, and such that the square
$$
\require{AMScd}
\begin{CD}
T^2L @>T\alpha_L >> TL \\
@V\mu_L VV @V\alpha_L VV\\
TL @>\alpha_L >> L
\end{CD}
$$
commutes.
Now $L$ is the limit of $GH$, so morphisms $\alpha_L$ from $TL$ to $L$ correspond to a unique cone
$\sigma_i \circ \alpha_L : TL \to GH_i$. (Conversely, a cone $\pi_i : TL\to GH_i$ gives such a morphism $\alpha_L$.)
Moreover, $\alpha_L$ is an algebra structure on $L$ making each $\sigma_i$ a
morphism of $T$-algebras if and only if the following diagram commutes:
$$
\begin{CD}
TL @>T\sigma_i >> TH_i \\
@V\alpha_L VV @V\alpha_{H_i}VV \\
L @>\sigma_i >> H_i.
\end{CD}
$$
Therefore in order for $\sigma_i$ to be a morphism of $T$-algebras, the
$T$-algebra structure on $L$ must be given by the morphism $\alpha_L : TL\to L$ corresponding to the cone $\alpha_{H_i}\circ T(\sigma_i)$. In particular, the $T$-algebra structure on $L$ making $(L,\sigma)$ into a cone in $T\Alg$ is unique, if it exists.
To see that it exists, we need to show that $\alpha_L$ satisfies the properties required to be a monad algebra.
For this, note that two morphisms to the limit $L$ are equal if and only if they are equal when you apply each $\sigma_i$.
So for the unit diagram, we need to show $\alpha_L \eta_L =1_L$. Applying $\sigma_i$, we get
$$\sigma_i\alpha_L\eta_L = \alpha_{H_i}(T\sigma_i)\eta_L = \alpha_{H_i}\eta_{H_i}\sigma_i = \sigma_i=\sigma_i 1_L,$$
with the first equality the definition of $\alpha_L$, the second equality is naturality
of $\eta$, and the third is the unit identity for $H_i$ as a $T$-algebra.
For the multiplication compatibility square, we need to show
$\alpha_L(T\alpha_L) = \alpha_L\mu_L$. Again, we apply $\sigma_i$. We get
$$
\begin{align}
\sigma_i\alpha_L(T\alpha_L)
&=
\alpha_{H_i}(T\sigma_i)(T\alpha_L)
\\
&=
\alpha_{H_i}(T(\sigma_i\alpha_L))
\\
&=
\alpha_{H_i}(T(\alpha_{H_i}(T\sigma_i)))
\\
&=
\alpha_{H_i}(T\alpha_{H_i})(T^2\sigma_i)
\\
&=
\alpha_{H_i}\mu_{H_i}(T^2\sigma_i)
\\
&=
\alpha_{H_i}(T\sigma_i)\mu_L
\\
&=
\sigma_i\alpha_L\mu_L,
\end{align}
$$
as required. I'm not going to add anything further explaining these steps in detail. Instead, I'll say that these come from a geometric observation about a cubical diagram we can draw.
The observation is that we know all the faces except the $L$ face commute, either by naturality, construction of $\alpha_L$, or the fact that $H_i$ is a $T$-algebra, so we can push the paths we care about around the cube across the faces that commute, and thereby demonstrate that they are equal.
Diagram drawn with tikzcd.
Thus $\alpha_L$ does define a $T$-algebra.
An aside on diagram chasing
Diagram chasing is a vaguely defined term. It refers to a style of argument, like the one just given, where to prove things, you draw a diagram of objects and morphisms, and then either move elements around the diagram, construct new morphisms, or use commutativity of the diagram (where it is commutative) to prove what you want to prove. I'm not sure there's a more specific definition I could give.
Back to the problem
We have now shown that there is a unique lift of a limiting cone to $T\Alg$ by producing a unique algebra structure on $L$. All that's left is to show that this lifted cone is also a limiting cone.
Since it's a limiting cone in $X$, this comes down to just showing that if $(A,\pi)$ is a cone to $H$ in $T\Alg$, then the induced map $p: A\to L$ in $X$ is actually a map of $T$-algebras. In other words, you need to show that
$$
\begin{CD}
TA @>Tp >> TL \\
@V\alpha_A VV @V\alpha_L VV\\
A @>p>> L \\
\end{CD}
$$
commutes. The solution is very similar to what we did above, so I'll leave this as an exercise.
