Python 2, 63 bytes

lambda a:[len(`map(max,v)`[1::3].strip('0'))for v in zip(*a),a]

An unnamed function accepting a list of lists of integers (in {0,1}) which returns a list of integers, [width,height].

Try it online!

Python 2,  63  55 bytes

-8 with help from Vincent (take the input matrix as a Numpy array)

lambda a:[len(`a.max(x)`[7::3].strip('0'))for x in 0,1]

An unnamed function accepting a 2-d Numpy array of integers (in {0,1}) which returns a list of integers, [width,height].

Try it online!

Non-Numpy version in 63 bytes (accepting a list of lists of integers in {0,1}):

lambda a:[len(`map(max,v)`[1::3].strip('0'))for v in zip(*a),a]

Try it online!

Given a matrix, a, for each (v) of the transpose, zip(*a), and a itself we find the height required (given the transpose this is the width).

Mapping max across v yields a list of zeros and ones, representing if each row of v contains any ones. The string representation of this list is found using backticks (`...`), this gives a string with a leading [, then the zeros and ones delimited by , . We slice this string starting a index one in steps of three using [1::3] getting us a string of only the zeros and ones, which allows us to use the string function strip to remove the outer zeros (strip('0')).

Given a matrix, a, for each (v) of the transpose, zip(*a), and a itself we find the height required (given the transpose this is the width).

Mapping max across v yields a list of zeros and ones, representing if each row of v contains any ones. The string representation of this list is found using backticks (`...`), this gives a string with a leading [, then the zeros and ones delimited by , (comma+space). We slice this string starting at index one in steps of three using [1::3] getting us a string of only the zeros and ones, which allows us to use the string function strip to remove the outer zeros (strip('0')).

How?

Given a matrix, a, for each (v) of the transpose, zip(*a), and a itself we find the height required (given the transpose this is the width).

Mapping max across v yields a list of zeros and ones, representing if each row of v contains any ones. The string representation of this list is found using backticks (`...`), this gives a string with a leading [, then the zeros and ones delimited by , . We slice this string starting a index one in steps of three using [1::3] getting us a string of only the zeros and ones, which allows us to use the string function strip to remove the outer zeros (strip('0')).

For example:

      a = [[0,0,0,0,0]           map(max,a)                    = [0,1,1]
          ,[0,1,0,0,0]          `map(max,a)`[1::3]             = '011'
          ,[0,0,0,1,0]]         `map(max,a)`[1::3].strip('0')  = '11'
                            len(`map(max,a)`[1::3].strip('0')) = 2

zip(*a) = [(0,0,0)         map(max,zip(*a))                    = [0,1,0,1,0]
          ,(0,1,0)        `map(max,zip(*a))`[1::3]             = '01010'
          ,(0,0,0)        `map(max,zip(*a))`[1::3].strip('0')  = '101'
          ,(0,0,1)    len(`map(max,zip(*a))`[1::3].strip('0')) = 3
          ,(0,0,0)]

    --> [len(`map(max,v)`[1::3].strip('0'))for v in zip(*a),a] = [3,2]