How to draw each section of the blue Bezier curves with different color?

Currently, you're joining the bezier curve together (using path = np.append(path, bezier). If you additionally create a list (e.g. line_collection) to hold each part of the bezier, you can plot them separately afterwards:

import numpy as np
import scipy.special
import matplotlib.pyplot as plt

def calc_bezier_path(control_points, n_points=100):
    """
    Compute bezier path (trajectory) given control points.
    :param control_points: (numpy array)
    :param n_points: (int) number of points in the trajectory
    :return: (numpy array)
    """
    traj = []
    for t in np.linspace(0, 1, n_points):
        traj.append(bezier(t, control_points))

    return np.array(traj)


def bernstein_poly(n, i, t):
    """
    Bernstein polynom.
    :param n: (int) polynom degree
    :param i: (int)
    :param t: (float)
    :return: (float)
    """
    return scipy.special.comb(n, i) * t ** i * (1 - t) ** (n - i)


def bezier(t, control_points):
    """
    Return one point on the bezier curve.
    :param t: (float) number in [0, 1]
    :param control_points: (numpy array)
    :return: (numpy array) Coordinates of the point
    """
    n = len(control_points) - 1
    return np.sum([bernstein_poly(n, i, t) * control_points[i] for i in range(n + 1)], axis=0)

def line_bezier(visx, visy, control, mod="nothing"):
    vis = np.column_stack((visx,visy))
    path_x, path_y = np.array([]),np.array([])
    setting = {"nothing":[len(vis)-2, 1, 1], "start":[len(vis)-1, 0, 0], "end":[len(vis)-1, 1, 0], "both":[len(vis), 0, -1]}
    epoch = setting[mod][0]
    start = setting[mod][1]
    end = setting[mod][2]
    
    ## create a new list to hold the line segments
    line_collection=[]
    if len(vis) > 2:
        current_control = vis[0]
        for x in range(epoch):
            if x != (epoch-1):
                for y in control:
                    if y == control[0]:
                        mid_control = [(vis[x+start,0]+(vis[x+(start+1),0]-vis[x+start,0])*y), (vis[x+start,1]+(vis[x+(start+1),1]-vis[x+start,1])*y)]
                        plt.annotate(f"M{x}", mid_control)
                        bezier_line = calc_bezier_path(np.array([current_control,vis[x+start], mid_control]))
                        path_x = np.append(path_x, bezier_line.T[0])
                        path_y = np.append(path_y, bezier_line.T[1])
                        
                        ## append the segment to the list
                        line_collection.append(bezier_line)
                        current_control = mid_control
                    else:
                        mid_control = [(vis[x+start,0]+(vis[x+(start+1),0]-vis[x+start,0])*y), (vis[x+start,1]+(vis[x+(start+1),1]-vis[x+start,1])*y)]
                        plt.annotate(f"M{x}", mid_control)
                        bezier_line = calc_bezier_path(np.array([current_control, mid_control]))
                        path_x = np.append(path_x, bezier_line.T[0])
                        path_y = np.append(path_y, bezier_line.T[1])
                        
                        ## append the segment to the list
                        line_collection.append(bezier_line)
                        current_control = mid_control
                        
            else:
                if mod == "end" or mod == "both":
                    bezier_line = calc_bezier_path(np.array([current_control, vis[x+(end+1)]]))
                else:
                    bezier_line = calc_bezier_path(np.array([current_control, vis[x+end], vis[x+(end+1)]]))
                path_x = np.append(path_x, bezier_line.T[0])
                path_y = np.append(path_y, bezier_line.T[1])
                ## append the segment to the list:
                line_collection.append(bezier_line)
                
    else:
        path_x, path_y = visx, visy
    return path_x, path_y, line_collection ## return the line_collection here


visx, visy = [1,2,10,15,20,25,21], [0,5,1,4,2,3,3]
control = [0.25,0.57]
path_x, path_y, line_collection = line_bezier(visx, visy, control,mod="end") ## init the line_collection

## iterate over list elements
for line_seg in line_collection:
    plt.plot(line_seg.T[0],line_seg.T[1],linestyle="solid",linewidth=5,alpha=.5,zorder=10)

# plt.plot(path_x, path_y) ## dont need to plot this now
plt.plot(visx, visy, "--o")
for xy in range(len(visx)):
    plt.annotate(f"P{xy}", [visx[xy], visy[xy]])
plt.show()

yields: