The mode in Dirichlet-Multinomial is
$$ \mathrm{Mode}(\pi_i) = \frac{\alpha_i + x_i - 1}{\sum_{j=1}^k (\alpha_j + x_j -1)} $$
Let $f(x_1,\dotsc,x_n)$ be the PDF of $\operatorname{Dir}(\alpha_1+1,\dotsc,\alpha_n+1)$. Then let
$$A=\log f(x_1,\dotsc,x_n)=\log(x_1^{\alpha_1}\dots x_n^{\alpha_n}) + C= \log(x_1^{\alpha_1})+\dots +\log(x_n^{\alpha_n}) + C$$ where $C$ is some constant.
We have this constraint on the variables $(x_1,\dotsc,x_n)$:
$$g(x_1,x_2,...,x_n) = \sum_i x_i =1 $$
Maximizing $A$ is the same as maximizing f. We introduce a Lagrange multiplier $\lambda$. Let the Lagrangian function be:
$$L(x_1,\dotsc,x_n,\lambda)= A-\lambda (g-c) = \log(x_1^{\alpha_1})+\dots +\log(x_n^{\alpha_n}) +C+ \lambda(x_1+\dots +x_n-1)$$ Taking the gradient of both sides gives:
$$d L(x_1,\dotsc,x_n) = \left({\alpha_1\over x_1} + \lambda\right)dx_1 + \dots \left({\alpha_n\over x_n} + \lambda\right)dx_n + (x_1+\dots +x_n-1)d\lambda$$
Solving for $dL=0$ gives $$\tag{1}x_i=-{\alpha_i \over \lambda} $$ and $$\sum_i x_i =1 \tag{2}$$ Apply the operator $\sum_i$ to $(1)$ taking into account $(2)$. This gives that $$\lambda = -\sum_i \alpha_i $$ which finally means that $$x_i=\frac{\alpha_i}{\sum_j \alpha_j} $$
The Dirichlet distribution represents an estimate of what categorical distribution produced some set of observations.
For example: If there's a scenario where there are three types of events:
event $1$ has been observed $5$ times
event $2$ has been observed $10$ times
event $3$ has been observed $7$ times,
then our "knowledge" about the probability distribution that produced those events is represented by the Dirichlet distribution $\operatorname{Dir}(5+1,10+1,7+1)$. It should almost be common sense that the most likely probabilities for events $1$, $2$ and $3$ should be $5\over5+10+7$, $10\over5+10+7$ and $7\over5+7+10$ respectively. Grouping them together as $(\frac5{22}, \frac{10}{22}, \frac{7}{22})$ then gives the mode of the Dirichlet distribution.
The above claim about the mode can be verified by solving a constrained optimisation problem. The optimisation needs to be constrained because the Dirichlet PDF is set to be zero outside those $(x_1, \dotsc, x_n)$ values for which $x_1+\dotsc+x_n=1$. This optimisation is easily done using Lagrange multipliers.
I've been trying to think about why the expression for the expected value is the same as the mode but the number of observations of each kind of event has increased by $1$. Proving it seems to be an exercise in integration. But I'd like to see an intuitive argument for that as well.
