It is better to use a date table that have consecutive dates in case dates in your table have gaps. I am wondering what's the role of the id column? Here is how I would do it without considering the id column.

select row_number()over(order by dt) as id
,dt as created_at
,cnt1+cnt2+cnt3+cnt4+cnt5 as cnt
from
(
    select
    date_table.dt
    ,lag(cnt,2,0)over(order by created_at asc) as cnt1
    ,lag(cnt,1,0)over(order by created_at asc) as cnt2
    ,isnull(cnt,0) cnt3
    ,lead(cnt,1,0)over(order by created_at asc) as cnt4
    ,lead(cnt,2,0)over(order by created_at asc) as cnt5
    from
    date_table left join
    (select created_at,count(*) as cnt from your_table group by created_at) c
on date_table.day = c.created_at
) T

Using window functions for this purpose is challenging because of the duplicate days. You can get the results using a join or correlated subquery:

select t.*,
       (select count(*)
        from t t2
        where t2 between t.created_at - interval 2 * '1 day' and 
                         t.created_at + interval 2 * '1 day'
       ) as cnt
from t;

EDIT:

You could use window functions by doing a cumulative sum by date and then joining back. This is, of course, a bit challenging because of holes in the dates. But, something like this:

with c as (
      select d.dte, count(t.created_at) as cnt,
             sum(count(t.created_at))) over (order by d.dte) as cumecnt
      from (select generate_series(min(created_at) - interval '2 day', 
                                   max(created_at) + interval '2 day', 
                                   '1 day')
            from t
           ) d(dte) left join
           on d.dte = t.created_at
     )
select t.*, cmax.cumecnt - cmin.cumecnt
from t join
     c cmin
     on t.created_at = cmin.dte + interval '2 day' join
     c cmax
     on t.created_at = cmax.dte - interval '2 day';