How, when and why does implicit changing of transaction isolation level happen?
How does the output illustrate implicit elevation of isolation level?

Sunil is technically correct, but it does sound a little confusing, I agree.
The output shows the session is blocked waiting to acquire a U lock. The definition of the READ COMMITTED isolation level is that the session will only encounter committed data. SQL Server honours this logical requirement under the default pessimistic (locking) implementation of read committed by holding shared locks just long enough to avoid seeing uncommitted data. These shared locks are normally quickly released (usually just before reading the next row).
Under optimistic (row-versioning) read committed (RCSI) SQL Server avoids reading uncommitted data by reading the last-committed version of the row at the time the statement started instead.
The sense Sunil is trying to convey is that taking U locks (instead of brief shared locks or reading versions) represents a (technical) escalation of isolation level (though not to any explicitly named level).
The effective isolation level in this case is not quite REPEATABLE READ because any U locks taken (and not converted to X locks) are released at the end of the statement. This is different from the behaviour of the UPDLOCK hint, which acquires and holds U locks (at least) until the end of the transaction. In addition, REPEATABLE READ generally acquires S locks (though this is strictly just an implementation detail).
Confusingly, the engine also takes U locks on the access method when identifying rows to update under default (locking) read-committed. This is a convenience to avoid a common deadlocking scenario without having to specify UPDLOCK explicitly. I apologise that this is so complicated, but there we are.

How to check for real isolation level &quot;jumpings&quot; in context of some statements?

There is nothing explicitly exposed in query plans to identify cases where the engine temporarily increases the effective isolation level. This might change in a future version of SQL Server. There may be indirect evidence in terms of locks taken, but this is rarely a convenient approach.

When to expect them and why do they occur?

Some of the occasions when internal escalation occurs are (somewhat) documented in Books Online. For example, Understanding Row Versioning-Based Isolation Levels says (among other things worth noting):

In a read-committed transaction using row versioning, the selection of rows to update is done using a blocking scan where an update (U) lock is taken on the data row as data values are read.

The general reason for temporary changes in effective isolation level changes is to avoid data corruption. A list of posts identifying some common cases follows:
Blocking Operators
Large Objects
Lookup with Prefetching
Cascading Referential Integrity
Other common cases (not a complete list):

Shared locks taken when the query processor verifies foreign key relationships.
Range locks taken when maintaining an indexed view referencing more than one table.
Range locks taken when maintaining an index with IGNORE_DUP_KEY.

Some of these behaviours may be documented in Books Online, somewhere, but there's no convenient single list that I am aware of.