This is spelled out in some detail here:
But the basic answer is that stored procedures have cached and reused query plans, and permissions checks are not necessary when objects owned by the stored procedure owner are accessed from a stored procedure. If valid ownership chains exist between the procedure and the object to be accessed, then only EXECUTE permission is checked before the stored procedure is started. Other permissions checks are skipped.
Could you come up with a step list about how a query (or SP) is processed after it is submitted?
The main thing to remember here is that all the queries in a batch are parsed and compiled before any of them starts to execute. Permissions are checked during execution. And it's even possible to reference a table in a query in such a way that permissions on that table are never actually checked.
If you watch an XE session like this:
CREATE EVENT SESSION [compiles] ON SERVER
ADD EVENT sqlserver.query_post_compilation_showplan(
WHERE ([sqlserver].[client_app_name]=N'Microsoft SQL Server Management Studio - Query'))
WITH (MAX_MEMORY=4096 KB,EVENT_RETENTION_MODE=ALLOW_SINGLE_EVENT_LOSS,MAX_DISPATCH_LATENCY=1 SECONDS)
And do something like
create table t(id int)
create table w(id int)
create user fred without login
grant select on w to fred
execute as user='fred'
select * from w
select * from t
where 1 = case when 1=1 then 1 else (select count(*) from t) end
You'll see that the query plans in the first batch are both compiled, then the first query runs, then the second fails. The third query succeeds without a permissions check.