Using Lgrange multipliers to optimize a function under constraints is a useful technique, although in the end, it provides additional insights and information. Sticking to the case of equality constraints, the problem
$$\max_{(x,y)} u(x,y) = x^{\alpha} y^{1-\alpha},\;\; \alpha \in (0,1)$$
$$\text {s.t.} \;\;w = p_xx + p_yy$$
can of course be transformed in an unconstrained problem by direct substitution:
$$\max_{y} u(x,y) = \left(\frac {w-yp_y}{p_x}\right)^{\alpha} y^{1-\alpha},\;\; \alpha \in (0,1)$$
But in general, direct substitution can produce cumbersome expressions (especially in dynamic problems), where an algebraic mistake will be easy to make. So the Lagrange method has an advantage here. Moreover, the Lagrange multiplier has a meaningful economic interpretation. In this approach, we define a new variable, say $\lambda$, and we form the "Lagrangean function"
$$\Lambda(x,y,\lambda) = x^{\alpha} y^{1-\alpha} + \lambda (w-p_xx-p_yy)$$
First, note that $\Lambda(x,y,\lambda)$ is equivalent to $u(x,y)$, since the added part to the right is identically zero. Now we maximize the Lagrangean with respect to the two variables and we obtain the first order conditions
$$\frac {\partial u}{\partial x} = \lambda p_x$$
$$\frac {\partial u}{\partial y} = \lambda p_y$$
Equating through $\lambda$, this provides quickly the fundamental relation
$$\frac {\partial u/\partial x} {\partial u/\partial y}= \frac {p_x}{p_y}$$
This optimal relation, together with the budget constraint, provide a two-equation system in two unknowns, and so provide the solution $(x^*, y^*)$ as a function of the exogenous parameters (the utility parameter $\alpha$, the prices $(p_x,p_y)$ and the given wealth $w$).
To determine the value of $\lambda$, multiply each first-order condition throughout by $x$ and $y$ respectively and then sum by sides to get
$$\frac {\partial u}{\partial x}x+\frac {\partial u}{\partial y}y = \lambda (p_xx+p_yy) = \lambda w$$
With utility homogeneous of degree one, as it is the case with Cobb-Douglas functions, we have that
$$\frac {\partial u}{\partial x}x+\frac {\partial u}{\partial y}y = u(x,y)$$
and so at the optimum bundle we have
$$u(x^*,y^*) =\lambda^*w$$
And this is how the Lagrange multiplier acquires an economically meaningful interpretation: its value is the marginal utility of wealth. Now, in the context of ordinal utility, marginal utility is not really meaningful (see also the discussion here). But the above procedure can be applied for example to a cost-minimization problem, where the Lagrange multiplier reflects the increase in total cost by a marginal increase in quantity produced, and so it is the Marginal Cost.