How to get the MSE of the node in the DecisionTreeRegressor of scikit-learn?

Nice question. You need tree_reg.tree_.impurity.

Short answer:

tree_reg = tree.DecisionTreeRegressor(max_depth=2)
tree_reg.fit(X_train, y_train)

extracted_MSEs = tree_reg.tree_.impurity # The Hidden magic is HERE

for idx, MSE in enumerate(tree_reg.tree_.impurity):
    print("Node {} has MSE {}".format(idx,MSE))

Node 0 has MSE 86.873403833
Node 1 has MSE 40.3211827171
Node 2 has MSE 25.6934820064
Node 3 has MSE 19.0053469592
Node 4 has MSE 74.6839429717
Node 5 has MSE 38.3057346817
Node 6 has MSE 39.6709615385

Long answer using the boston dataset with visual output:

import pandas as pd
import numpy as np
from sklearn import ensemble, model_selection, metrics, datasets, tree
import graphviz

house_prices = datasets.load_boston()

X_train, X_test, y_train, y_test = model_selection.train_test_split(
    pd.DataFrame(house_prices.data, columns=house_prices.feature_names),
    pd.Series(house_prices.target, name="med_price"),
    test_size=0.20, random_state=42)

tree_reg = tree.DecisionTreeRegressor(max_depth=2)
tree_reg.fit(X_train, y_train)

extracted_MSEs = tree_reg.tree_.impurity # YOU NEED THIS 
print(extracted_MSEs)
#[86.87340383 40.32118272 25.69348201 19.00534696 74.68394297 38.30573468 39.67096154]

# Compare visually
dot_data = tree.export_graphviz(tree_reg, out_file=None, feature_names=X_train.columns)
graph = graphviz.Source(dot_data)

#this will create an boston.pdf file with the rule path
graph.render("boston")

Compare MSE values with visual Output: