Currently, I am reading his annual letter. In his letter, he always says that his fund performance is likely to be better in a bear market than in a bull market, but I don't know why. Could someone explain this to me?
Buffett is a value investor. His goal is to buy good companies when the market is overly worried and prices them below intrinsic value. When the market is highly priced it is much more difficult for him to find things that he thinks are at an attractive price. When people are very worried and the market has crashed, stocks are then priced below their intrinsic value and he can use the cash he keeps in the company to make attractive purchases.
Remember that Buffett is not concerned with the ups and downs of the price of Berkshire Hathaway stock, he is concerned with the economic value of the assets that the company owns. So if all stock prices crash and he can buy things that are at bargain prices, he is happy no matter what Berkshire stock price does in the short run.
One consequence of value investing is that because you are buying assets at bargain prices, the total value of your assets drops less in a bear market than the highly priced stuff that drives the major indexes.
From the letter you link:
Our performance, relatively, is likely to be better in a bear market than in a bull market so that deductions made from the above results should be tempered by the fact that it was the type of year when we should have done relatively well. In a year when the general market had a substantial advance I would be well satisfied to match the advance of the Averages.
Putting those two sentences together, the word relatively means that his funds perform better than the market in bear markets and perform about the same as the overall market in bull markets. It does not mean that absolute performance is better in bear markets than bull markets.
Later on he states
This policy should lead to superior results in bear markets and average performance in bull markets.
Warren Buffet and Berkshire Hathaway took a 50% loss in each of the last two bear markets. His stock even lost 10% in 2015 when the S&P lost 8%.
He doesn't have a track record to support the claim that his stock performs relatively better in a bear market, so perhaps it's best to take his letter with a grain of salt.
Edit: As one commenter points out, Mr. Buffett is comparing the book performance of his fund to the market performance of an index. That is an apples to oranges comparison. It's deceptive at best.
To understand his comments about bear-market performance it's important to take them in context. (My research method was Crtl+F: bear; read around the highlights. This is not a complete survey of 60+ years of letters.)
In his earlier letters, statements about bull market performance are always made in reference to Buffet's belief that many of BH's current holdings are in undervalued securities. Ex:
To the extent possible, therefore, I am attempting to create my own work-outs by acquiring large positions in several undervalued securities. Such a policy should lead to the fulfillment of my earlier forecast – an above average performance in a bear market. It is on this basis that I hope to be judged (p 6; emphasis mine).
Similar statements are made throughout the earlier letters, along with this interesting note:
In a year when the general market had a substantial advance I would be well satisfied to match the advance of the Averages (p 6).
So to your question of why BH fund performance is likely to be better in a bear market than in a bull market, I believe the implicit assertion is that undervalued securities are more resilient in a bear market (presumably because they don't have as far to fall, and are also less likely to be subject to a bubble). Buffet is also explicitly asserting that when facing a choice to either (a) position BH to weather a possible downturn or (b) position BH to enjoy a bullish stock that is outpacing the market, he would choose the former over the later.
As to your assertion that he always says this, I can find no reference to bear market's in the letters past 1960.