Why Sum in database query giving NULL

All arithmetic or logical operations involving NULL yield NULL. For example:

SELECT 1 + NULL -- NULL

You must convert NULL to zeros before you can + them:

SELECT
    COALESCE(SUM(January), 0) +
    COALESCE(SUM(February) , 0) +
    ...

It is also possible to add the columns first and then calculate the sum:

SELECT SUM(
    COALESCE(January, 0) +
    COALESCE(February, 0) +
)

Be advised that (i) SUM skips NULL values (ii) returns NULL instead of 0 if all values are NULL:

SELECT SUM(a) FROM (VALUES
    (1),
    (2),
    (NULL)
) AS v(a) -- returns 3 instead of NULL

It will return NULL if all values encountered were NULL:

SELECT SUM(a) FROM (VALUES
    (CAST(NULL AS INT)),
    (NULL),
    (NULL)
) AS v(a) -- returns NULL instead of 0

use coalesce function to convert null to 0 then use sum

Select Sum(coalesce(January,0)) + Sum (coalesce(February,0)) ..... (I skipped the rest 10 months)
from Expense

That because you have NULL values, you can use Case, Coalesce or IIF:

Select SUM(IIF(Col IS NULL, 0, Col))

Select SUM(CASE WHEN Col IS NULL THEN 0 ELSE Col END)

Select COALESCE(Sum(Col), 0)

Any arithmetic function will return null if there is at least one null value in the given column. That's why you should use functions like coalesce or isNull (MSSQL), NVL (Oracle).

You can use ISNULL(SUM(January),0).

Because null + value is always null and in your sample some months sums are null, you can avoid this by adding ISNULL

Select isnull(Sum(January),0) + 
    isnull(Sum(February),0) 
--..... (I skipped the rest 10 months)
from Expense 

Alternatively you can use below way:

Select Sum(
    isnull(January,0) + 
    isnull(February,0) 
    )
--..... (I skipped the rest 10 months)
from Expense