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Why does the NASA Flight Director say, "Lock the doors.", when realizing that disaster had struck for Space Shuttle Columbia, back in 2003?

Does he mean lock the doors on the space shuttle that's falling down, or does he mean lock the doors in the NASA building, in order to begin some internal investigation, and nobody is allowed to leave, as a matter of policy?

For reference, it happens at about 3:14 in this YouTube video:

This is indeed part of the procedure that is invoked when a contingency has occurred. It is part of Standard Operation Procedure 2.8 - JSC Contingency Plan, which can be found in the Shuttle Flight Control Operations Handbook (link to 538-page pdf - referenced here) on page 2.8-1. It provides the steps to be taken to secure all data for future investigations (including any trash!).

It doesn't literally say to lock the doors, but item 11 on the checklist (Table 2.8-2) says:

Ensure area security: Personnel manning MCC positions listed in table 2.8-3 will assume responsibility for area security.

Which may be interpreted as "locking the doors".

Does he mean lock the doors in the NASA building, in order to begin some internal investigation, and nobody is allowed to leave, as a matter of policy?

Yes, this. It's part of a standard procedure to ensure evidence is preserved for the investigation. It's to prevent people entering as well as leaving.

While this is much granier than the BBC snippet, it does contain the audio overlay and is the same video the BBC clip has, but contains the full event cycle. "Lock the doors" happens around 12:24

Key parts of the full audio (time in the video first)

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  13:41 - OK all flight controllers on the Flight [Director] loop, we need to kick off the FCOH [Flight Control Operations Handbook] contingency plan procedure, FCOH checklist, page 2.8-5.

  
  

  
  
  

  From the link in Ludo's answer we see that 2.8-5 is

  
  
  

  
  

  Notify (or verify the notification of) MCC and other JSC support areas.

  
  

  
  


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  15:10 - Ok, all controllers, on Page 9 of the FCOH procedure, you need to make sure you step through the actions required in Step 20. That's for your workstation logs, display printouts -- there's a whole list of data collection items we need to make sure we log through

  
  

  
  
  

  And Step 20 is

  
  
  

  
  

  20. Collect the following, as appropriate, for subsequent
      turn-in or possible use by the Board of Investigation.
  
  

  
  


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  16:04 - Folks, listening in on the Flight loop -- No phone calls off site, outside of this room, our discussions are on these loops; the recorded DVIS loops only. No phone calls, no transmissions anywhere into or out.

  
  

  
  
  

  This is the purpose of "lock the doors". The fact that we have this video of Houston and what went on in its entirety is critical preservation of evidence. Taking time to talk offsite, even for official business, could have meant vital data (like thought processes and decision making) was lost.

  
  

Because the Handbook told him to was assuredly the proximate cause for him saying the phrase, and evidence preservation is undoubtedly a good reason the manual says to do so, but--so much as a movie and my memory can be trusted--in The Right Stuff, during Mercury, higher ups overruled the Flight Director on an important call during an anomaly. I believe the historical and ultimate reason they lock the doors is to ensure Mission Control staff is allowed to do their jobs focused on the mission and astronauts' lives, and not be interfered with by people seeking a particular PR or political outcome.