Why did annual consumer price inflation in May (excluding food and energy) reach the highest level in 29 years?

As usual with prices, it is because of supply and demand. Demand has been going up due to the stimulus. Supply has been "constricted." Not clear if that means it decreased or simply did not increase enough to prevent the demand increase from bringing us such large price increases. If they both increase, but demand shifts more than supply, then prices rise.

See Risks Rise to Both the Fed’s Inflation and Employment Goals: Federal Reserve bets supply-driven inflation will be transitory—but that depends on inflation expectations staying low by Greg Ip of The WSJ. Excerpts:

"unemployment is still around 6% and payrolls are 7.6 million short of pre-pandemic levels. Yet annual consumer price inflation excluding food and energy reached 3.8% in May, the highest in 29 years, and ran at an 8% annualized rate in the past three months. Annual inflation will likely stay above 3% through the rest of this year.

Economists expected some price pressure as the economy reopened, but not this much. What happened? First, demand came back faster than expected thanks in part to two rounds of fiscal stimulus. Last September the Fed projected economic growth this year of 4%. By March, they had raised that to 6.5%.

Second, supply has been constricted by a litany of freak events. Pandemic-induced cuts to production and shifts in demand patterns, a container ship jamming the Suez Canal, a fire at a Japanese microchip factory and a winter storm in Texas created shortages of everything from used cars and semiconductors to chicken wings and lumber. Most strikingly, many workers who left the labor force last year have yet to return, putting upward pressure on wages."

"used-cars sales this year are roughly level with 2019, according to Cox Automotive. But supply has been held back by people hanging on to leased vehicles longer, and rental-car agencies having fewer vehicles to sell. That has pushed prices up 18% in the past two months according to the Labor Department, contributing about 0.5 percentage point to inflation."

"Enhanced unemployment benefits, the virus and changed priorities are keeping them [workers] out."

"The Fed can’t do anything about supply shocks." [chairman Powell said so recently]

"The inflation effect of supply shocks is transitory so long as the public doesn’t expect permanently higher future inflation. Higher expectations can change wage and price-setting behavior and thus become self-fulfilling." [if workers expect higher inflation they will ask for or seek higher wages before the higher prices come-that means higher resource prices which reduce supply and raise product prices]

"Back in the 1970s, expectations were poorly anchored so oil shocks in 1973 and 1979 fed into broader inflation, forcing the Fed to raise interest rates sharply, causing deep recessions.

But since the 1990s, inflation expectations have been relatively stable around 2% in most advanced countries"

"What should the Fed do? It won’t know for months whether the rise in inflation is more than transitory"